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

Solution

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

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

Solution steps

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

Subtract

cos^{2} (x)

from both sides

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

Factor

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

(sin (2 x) + cos (x)) (sin (2 x) - cos (x)) = 0

Solving each part separately

sin (2 x) + cos (x) = 0 or sin (2 x) - cos (x) = 0

sin (2 x) + cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

sin (2 x) - cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Combine all the solutions

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

- 2 sin (2 x) sin (x) = sin (2 x)

2 sin (θ) - 2 = 0, 0 \leq θ \leq 2 π

sin (θ) - 0.2 cos (θ) = \frac{6.25}{9.8}

3 cos (θ) = 8 tan (θ)

tan (\frac{θ}{4}) + 3 = 0