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

Solution

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

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

Solution steps

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

Subtract

1 + sin (2 x) - sin (x)

from both sides

cos (2 x) + cos (x) - 1 - sin (2 x) + sin (x) = 0

Rewrite using trig identities

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

Factor

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

(1 - 2 sin (x)) (cos (x) + sin (x)) = 0

Solving each part separately

1 - 2 sin (x) = 0 or cos (x) + sin (x) = 0

1 - 2 sin (x) = 0 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

cos (x) + sin (x) = 0 : x = \frac{3 π}{4} + πn

Combine all the solutions

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

- sin (θ) + 2 sin^{2} (θ) = 0

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

2 tan (x) - 3 sin (x) = 0

sin (x) + 5 = 4

- 9 cos (x) = 0