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

Solution

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

x = 1.17809 \dots + 2 πn, x = π - 1.17809 \dots + 2 πn, x = - 1.17809 \dots + 2 πn, x = π + 1.17809 \dots + 2 πn

Solution steps

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

Subtract

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

from both sides

cos^{2} (x) - sin^{2} (x) + \frac{1}{2} = 0

Simplify

cos^{2} (x) - sin^{2} (x) + \frac{1}{2} : \frac{2 cos ^{2} ( x ) - 2 sin ^{2} ( x ) + 1}{2}

\frac{2 cos ^{2} ( x ) - 2 sin ^{2} ( x ) + 1}{2} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

2 cos^{2} (x) - 2 sin^{2} (x) + 1 = 0

Rewrite using trig identities

1 + 2 - 2 sin^{2} (x) 2 = 0

Solve by substitution

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

sin (x) = \frac{2 + 2}{2} : x = arcsin (\frac{2 + 2}{2}) + 2 πn, x = π - arcsin (\frac{2 + 2}{2}) + 2 πn

sin (x) = - \frac{2 + 2}{2} : x = arcsin (- \frac{2 + 2}{2}) + 2 πn, x = π + arcsin (\frac{2 + 2}{2}) + 2 πn

Combine all the solutions

x = arcsin (\frac{2 + 2}{2}) + 2 πn, x = π - arcsin (\frac{2 + 2}{2}) + 2 πn, x = arcsin (- \frac{2 + 2}{2}) + 2 πn, x = π + arcsin (\frac{2 + 2}{2}) + 2 πn

Show solutions in decimal form

x = 1.17809 \dots + 2 πn, x = π - 1.17809 \dots + 2 πn, x = - 1.17809 \dots + 2 πn, x = π + 1.17809 \dots + 2 πn

cos (2 θ) - sin (θ) = 0

tan (θ) = - \frac{2 3}{3} sin (θ)

3 cos (x) tan (x) - cos (x) = 0

4 sin^{2} (θ) = 3

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