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2-2cos^2(x/2)=2cos^2(x)

Solution

2 - 2 cos^{2} (\frac{x}{2}) = 2 cos^{2} (x)

Solution

x = π + 4 πn, x = 3 π + 4 πn, x = \frac{π}{3} + 4 πn, x = \frac{11 π}{3} + 4 πn, x = \frac{5 π}{3} + 4 πn, x = \frac{7 π}{3} + 4 πn

Degrees

x = 18 0^{\circ} + 72 0^{\circ} n, x = 54 0^{\circ} + 72 0^{\circ} n, x = 6 0^{\circ} + 72 0^{\circ} n, x = 66 0^{\circ} + 72 0^{\circ} n, x = 30 0^{\circ} + 72 0^{\circ} n, x = 42 0^{\circ} + 72 0^{\circ} n

Solution steps

2 - 2 cos^{2} (\frac{x}{2}) = 2 cos^{2} (x)

Subtract

2 cos^{2} (x)

from both sides

2 - 2 cos^{2} (\frac{x}{2}) - 2 cos^{2} (x) = 0

Let:

u = \frac{x}{2}

2 - 2 cos^{2} (u) - 2 cos^{2} (2 u) = 0

Rewrite using trig identities

2 - 2 cos^{2} (2 u) - 2 cos^{2} (u)

Use the Double Angle identity:

cos (2 x) = 2 cos^{2} (x) - 1

= 2 - 2 (2 cos^{2} (u) - 1)^{2} - 2 cos^{2} (u)

Simplify

2 - 2 (2 cos^{2} (u) - 1)^{2} - 2 cos^{2} (u) : 6 cos^{2} (u) - 8 cos^{4} (u)

2 - 2 (2 cos^{2} (u) - 1)^{2} - 2 cos^{2} (u)

(2 cos^{2} (u) - 1)^{2} : 4 cos^{4} (u) - 4 cos^{2} (u) + 1

Apply Perfect Square Formula:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = 2 cos^{2} (u), b = 1

= (2 cos^{2} (u))^{2} - 2 \cdot 2 cos^{2} (u) \cdot 1 + 1^{2}

Simplify

(2 cos^{2} (u))^{2} - 2 \cdot 2 cos^{2} (u) \cdot 1 + 1^{2} : 4 cos^{4} (u) - 4 cos^{2} (u) + 1

(2 cos^{2} (u))^{2} - 2 \cdot 2 cos^{2} (u) \cdot 1 + 1^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= (2 cos^{2} (u))^{2} - 2 \cdot 2 \cdot 1 \cdot cos^{2} (u) + 1

(2 cos^{2} (u))^{2} = 4 cos^{4} (u)

(2 cos^{2} (u))^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{2} (cos^{2} (u))^{2}

(cos^{2} (u))^{2} : cos^{4} (u)

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= cos^{2 \cdot 2} (u)

Multiply the numbers:

2 \cdot 2 = 4

= cos^{4} (u)

= 2^{2} cos^{4} (u)

2^{2} = 4

= 4 cos^{4} (u)

2 \cdot 2 cos^{2} (u) \cdot 1 = 4 cos^{2} (u)

2 \cdot 2 cos^{2} (u) \cdot 1

Multiply the numbers:

2 \cdot 2 \cdot 1 = 4

= 4 cos^{2} (u)

= 4 cos^{4} (u) - 4 cos^{2} (u) + 1

= 4 cos^{4} (u) - 4 cos^{2} (u) + 1

= 2 - 2 (4 cos^{4} (u) - 4 cos^{2} (u) + 1) - 2 cos^{2} (u)

Expand

- 2 (4 cos^{4} (u) - 4 cos^{2} (u) + 1) : - 8 cos^{4} (u) + 8 cos^{2} (u) - 2

- 2 (4 cos^{4} (u) - 4 cos^{2} (u) + 1)

Distribute parentheses

= (- 2) \cdot 4 cos^{4} (u) + (- 2) (- 4 cos^{2} (u)) + (- 2) \cdot 1

Apply minus-plus rules

+ (- a) = - a, (- a) (- b) = ab

= - 2 \cdot 4 cos^{4} (u) + 2 \cdot 4 cos^{2} (u) - 2 \cdot 1

Simplify

- 2 \cdot 4 cos^{4} (u) + 2 \cdot 4 cos^{2} (u) - 2 \cdot 1 : - 8 cos^{4} (u) + 8 cos^{2} (u) - 2

- 2 \cdot 4 cos^{4} (u) + 2 \cdot 4 cos^{2} (u) - 2 \cdot 1

Multiply the numbers:

2 \cdot 4 = 8

= - 8 cos^{4} (u) + 8 cos^{2} (u) - 2 \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= - 8 cos^{4} (u) + 8 cos^{2} (u) - 2

= - 8 cos^{4} (u) + 8 cos^{2} (u) - 2

= 2 - 8 cos^{4} (u) + 8 cos^{2} (u) - 2 - 2 cos^{2} (u)

Simplify

2 - 8 cos^{4} (u) + 8 cos^{2} (u) - 2 - 2 cos^{2} (u) : 6 cos^{2} (u) - 8 cos^{4} (u)

2 - 8 cos^{4} (u) + 8 cos^{2} (u) - 2 - 2 cos^{2} (u)

Group like terms

= - 8 cos^{4} (u) + 8 cos^{2} (u) - 2 cos^{2} (u) + 2 - 2

Add similar elements:

8 cos^{2} (u) - 2 cos^{2} (u) = 6 cos^{2} (u)

= - 8 cos^{4} (u) + 6 cos^{2} (u) + 2 - 2

2 - 2 = 0

= 6 cos^{2} (u) - 8 cos^{4} (u)

= 6 cos^{2} (u) - 8 cos^{4} (u)

= 6 cos^{2} (u) - 8 cos^{4} (u)

6 cos^{2} (u) - 8 cos^{4} (u) = 0

Solve by substitution

6 cos^{2} (u) - 8 cos^{4} (u) = 0

Let:

cos (u) = u

6 u^{2} - 8 u^{4} = 0

6 u^{2} - 8 u^{4} = 0 : u = 0, u = \frac{3}{2}, u = - \frac{3}{2}

6 u^{2} - 8 u^{4} = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

- 8 u^{4} + 6 u^{2} = 0

Rewrite the equation with

v = u^{2}

and

v^{2} = u^{4}

- 8 v^{2} + 6 v = 0

Solve

- 8 v^{2} + 6 v = 0 : v = 0, v = \frac{3}{4}

- 8 v^{2} + 6 v = 0

Solve with the quadratic formula

- 8 v^{2} + 6 v = 0

Quadratic Equation Formula:

For

a = - 8, b = 6, c = 0

v_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 ( - 8 ) \cdot 0}{2 ( - 8 )}

v_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 ( - 8 ) \cdot 0}{2 ( - 8 )}

6^{2} - 4 (- 8) \cdot 0 = 6

6^{2} - 4 (- 8) \cdot 0

Apply rule

- (- a) = a

= 6^{2} + 4 \cdot 8 \cdot 0

Apply rule

0 \cdot a = 0

= 6^{2} + 0

6^{2} + 0 = 6^{2}

= 6^{2}

Apply radical rule:

assuming

a \geq 0

= 6

v_{1, 2} = \frac{- 6 \pm 6}{2 ( - 8 )}

Separate the solutions

v_{1} = \frac{- 6 + 6}{2 ( - 8 )}, v_{2} = \frac{- 6 - 6}{2 ( - 8 )}

v = \frac{- 6 + 6}{2 ( - 8 )} : 0

\frac{- 6 + 6}{2 ( - 8 )}

Remove parentheses:

(- a) = - a

= \frac{- 6 + 6}{- 2 \cdot 8}

Add/Subtract the numbers:

- 6 + 6 = 0

= \frac{0}{- 2 \cdot 8}

Multiply the numbers:

2 \cdot 8 = 16

= \frac{0}{- 16}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{0}{16}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= - 0

= 0

v = \frac{- 6 - 6}{2 ( - 8 )} : \frac{3}{4}

\frac{- 6 - 6}{2 ( - 8 )}

Remove parentheses:

(- a) = - a

= \frac{- 6 - 6}{- 2 \cdot 8}

Subtract the numbers:

- 6 - 6 = - 12

= \frac{- 12}{- 2 \cdot 8}

Multiply the numbers:

2 \cdot 8 = 16

= \frac{- 12}{- 16}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{12}{16}

Cancel the common factor:

4

= \frac{3}{4}

The solutions to the quadratic equation are:

v = 0, v = \frac{3}{4}

v = 0, v = \frac{3}{4}

Substitute back

v = u^{2},

solve for

u

Solve

u^{2} = 0 : u = 0

u^{2} = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

u = 0

Solve

u^{2} = \frac{3}{4} : u = \frac{3}{2}, u = - \frac{3}{2}

u^{2} = \frac{3}{4}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{3}{4}, u = - \frac{3}{4}

\frac{3}{4} = \frac{3}{2}

\frac{3}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{3}{2}

- \frac{3}{4} = - \frac{3}{2}

- \frac{3}{4}

Simplify

\frac{3}{4} : \frac{3}{2}

\frac{3}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{3}{2}

= - \frac{3}{2}

u = \frac{3}{2}, u = - \frac{3}{2}

The solutions are

u = 0, u = \frac{3}{2}, u = - \frac{3}{2}

Substitute back

u = cos (u)

cos (u) = 0, cos (u) = \frac{3}{2}, cos (u) = - \frac{3}{2}

cos (u) = 0, cos (u) = \frac{3}{2}, cos (u) = - \frac{3}{2}

cos (u) = 0 : u = \frac{π}{2} + 2 πn, u = \frac{3 π}{2} + 2 πn

cos (u) = 0

General solutions for

cos (u) = 0

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

u = \frac{π}{2} + 2 πn, u = \frac{3 π}{2} + 2 πn

u = \frac{π}{2} + 2 πn, u = \frac{3 π}{2} + 2 πn

cos (u) = \frac{3}{2} : u = \frac{π}{6} + 2 πn, u = \frac{11 π}{6} + 2 πn

cos (u) = \frac{3}{2}

General solutions for

cos (u) = \frac{3}{2}

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

u = \frac{π}{6} + 2 πn, u = \frac{11 π}{6} + 2 πn

u = \frac{π}{6} + 2 πn, u = \frac{11 π}{6} + 2 πn

cos (u) = - \frac{3}{2} : u = \frac{5 π}{6} + 2 πn, u = \frac{7 π}{6} + 2 πn

cos (u) = - \frac{3}{2}

General solutions for

cos (u) = - \frac{3}{2}

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

u = \frac{5 π}{6} + 2 πn, u = \frac{7 π}{6} + 2 πn

u = \frac{5 π}{6} + 2 πn, u = \frac{7 π}{6} + 2 πn

Combine all the solutions

u = \frac{π}{2} + 2 πn, u = \frac{3 π}{2} + 2 πn, u = \frac{π}{6} + 2 πn, u = \frac{11 π}{6} + 2 πn, u = \frac{5 π}{6} + 2 πn, u = \frac{7 π}{6} + 2 πn

Substitute back

u = \frac{x}{2}

\frac{x}{2} = \frac{π}{2} + 2 πn : x = π + 4 πn

\frac{x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{π}{2} + 2 \cdot 2 πn : π + 4 πn

2 \cdot \frac{π}{2} + 2 \cdot 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{π 2}{2}

Cancel the common factor:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= π + 4 πn

x = π + 4 πn

x = π + 4 πn

x = π + 4 πn

\frac{x}{2} = \frac{3 π}{2} + 2 πn : x = 3 π + 4 πn

\frac{x}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn : 3 π + 4 πn

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

2 \cdot \frac{3 π}{2} = 3 π

2 \cdot \frac{3 π}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{3 π 2}{2}

Cancel the common factor:

2

= 3 π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= 3 π + 4 πn

x = 3 π + 4 πn

x = 3 π + 4 πn

x = 3 π + 4 πn

\frac{x}{2} = \frac{π}{6} + 2 πn : x = \frac{π}{3} + 4 πn

\frac{x}{2} = \frac{π}{6} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{π}{6} + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot \frac{π}{6} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} = 2 \cdot \frac{π}{6} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{π}{6} + 2 \cdot 2 πn : \frac{π}{3} + 4 πn

2 \cdot \frac{π}{6} + 2 \cdot 2 πn

2 \cdot \frac{π}{6} = \frac{π}{3}

2 \cdot \frac{π}{6}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{π 2}{6}

Cancel the common factor:

2

= \frac{π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{π}{3} + 4 πn

x = \frac{π}{3} + 4 πn

x = \frac{π}{3} + 4 πn

x = \frac{π}{3} + 4 πn

\frac{x}{2} = \frac{11 π}{6} + 2 πn : x = \frac{11 π}{3} + 4 πn

\frac{x}{2} = \frac{11 π}{6} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{11 π}{6} + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot \frac{11 π}{6} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} = 2 \cdot \frac{11 π}{6} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{11 π}{6} + 2 \cdot 2 πn : \frac{11 π}{3} + 4 πn

2 \cdot \frac{11 π}{6} + 2 \cdot 2 πn

2 \cdot \frac{11 π}{6} = \frac{11 π}{3}

2 \cdot \frac{11 π}{6}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{11 π 2}{6}

Multiply the numbers:

11 \cdot 2 = 22

= \frac{22 π}{6}

Cancel the common factor:

2

= \frac{11 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{11 π}{3} + 4 πn

x = \frac{11 π}{3} + 4 πn

x = \frac{11 π}{3} + 4 πn

x = \frac{11 π}{3} + 4 πn

\frac{x}{2} = \frac{5 π}{6} + 2 πn : x = \frac{5 π}{3} + 4 πn

\frac{x}{2} = \frac{5 π}{6} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{5 π}{6} + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot \frac{5 π}{6} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} = 2 \cdot \frac{5 π}{6} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{5 π}{6} + 2 \cdot 2 πn : \frac{5 π}{3} + 4 πn

2 \cdot \frac{5 π}{6} + 2 \cdot 2 πn

2 \cdot \frac{5 π}{6} = \frac{5 π}{3}

2 \cdot \frac{5 π}{6}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{5 π 2}{6}

Multiply the numbers:

5 \cdot 2 = 10

= \frac{10 π}{6}

Cancel the common factor:

2

= \frac{5 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{5 π}{3} + 4 πn

x = \frac{5 π}{3} + 4 πn

x = \frac{5 π}{3} + 4 πn

x = \frac{5 π}{3} + 4 πn

\frac{x}{2} = \frac{7 π}{6} + 2 πn : x = \frac{7 π}{3} + 4 πn

\frac{x}{2} = \frac{7 π}{6} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{7 π}{6} + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} = 2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn : \frac{7 π}{3} + 4 πn

2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn

2 \cdot \frac{7 π}{6} = \frac{7 π}{3}

2 \cdot \frac{7 π}{6}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{7 π 2}{6}

Multiply the numbers:

7 \cdot 2 = 14

= \frac{14 π}{6}

Cancel the common factor:

2

= \frac{7 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{7 π}{3} + 4 πn

x = \frac{7 π}{3} + 4 πn

x = \frac{7 π}{3} + 4 πn

x = \frac{7 π}{3} + 4 πn

x = π + 4 πn, x = 3 π + 4 πn, x = \frac{π}{3} + 4 πn, x = \frac{11 π}{3} + 4 πn, x = \frac{5 π}{3} + 4 πn, x = \frac{7 π}{3} + 4 πn

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