What is the general solution for cos^2(x/2)-sin^2(x/2)=sin(x) ?

The general solution for cos^2(x/2)-sin^2(x/2)=sin(x) is x= pi/4+pin

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

Solution

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

Solution

x = \frac{π}{4} + πn

Degrees

x = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

sin (x)

from both sides

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

= - sin (x) + cos (2 \cdot \frac{x}{2})

Multiply

2 \cdot \frac{x}{2} : x

2 \cdot \frac{x}{2}

Multiply fractions:

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

= \frac{x \cdot 2}{2}

Cancel the common factor:

2

= x

= - sin (x) + cos (x)

Divide both sides by

cos (x), cos (x) \neq = 0

\frac{cos ( x ) - sin ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

1 - \frac{sin ( x )}{cos ( x )} = 0

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

1 - tan (x) = 0

1 - tan (x) = 0

Move

1

to the right side

1 - tan (x) = 0

Subtract

1

from both sides

1 - tan (x) - 1 = 0 - 1

Simplify

- tan (x) = - 1

- tan (x) = - 1

Divide both sides by

- 1

- tan (x) = - 1

Divide both sides by

- 1

\frac{- tan ( x )}{- 1} = \frac{- 1}{- 1}

Simplify

tan (x) = 1

tan (x) = 1

General solutions for

tan (x) = 1

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

Popular Examples

tan(x)+sec(x)=-1

tan (x) + sec (x) = - 1

2sin^2(θ)-cos(θ)-1=0

2 sin^{2} (θ) - cos (θ) - 1 = 0

cos(x)+1=2sin^2(x)

cos (x) + 1 = 2 sin^{2} (x)

8sec^2(x/2)=4

8 sec^{2} (\frac{x}{2}) = 4

6cos^2(x)+cos(x)=1

6 cos^{2} (x) + cos (x) = 1

Frequently Asked Questions (FAQ)

What is the general solution for cos^2(x/2)-sin^2(x/2)=sin(x) ?
The general solution for cos^2(x/2)-sin^2(x/2)=sin(x) is x= pi/4+pin