What is the general solution for sqrt(2)cos^2(x)=cos^2(x) ?

The general solution for sqrt(2)cos^2(x)=cos^2(x) is x= pi/2+2pin,x=(3pi)/2+2pin

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sqrt(2)cos^2(x)=cos^2(x)

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

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

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

cos (x) = u

2 u^{2} = u^{2}

2 u^{2} = u^{2} : u = 0

2 u^{2} = u^{2}

Move

u^{2}

to the left side

2 u^{2} = u^{2}

Subtract

u^{2}

from both sides

2 u^{2} - u^{2} = u^{2} - u^{2}

Simplify

2 u^{2} - u^{2} = 0

2 u^{2} - u^{2} = 0

Factor

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

2 u^{2} - u^{2}

Factor out common term

u^{2}

= u^{2} (2 - 1)

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

Divide both sides by

2 - 1

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

Divide both sides by

2 - 1

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

Divide both sides by

2 - 1

\frac{( 2 - 1 ) u ^{2}}{2 - 1} = \frac{0}{2 - 1}

Simplify

u^{2} = 0

u^{2} = 0

Apply rule

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

u = 0

Substitute back

u = cos (x)

cos (x) = 0

cos (x) = 0

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

cos (x) = 0

General solutions for

cos (x) = 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}

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

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

Combine all the solutions

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

sin (\frac{π x}{2}) = 0

sin (x) = \frac{1}{2} 2

8 cos (2 x) = 2 sin (x) + 7

cos (θ) = 0.5332, θ

sin (k) = 0.5

What is the general solution for sqrt(2)cos^2(x)=cos^2(x) ?
The general solution for sqrt(2)cos^2(x)=cos^2(x) is x= pi/2+2pin,x=(3pi)/2+2pin