What is the general solution for (1-sin(x))(1+cos(x))=cos^2(x) ?

The general solution for (1-sin(x))(1+cos(x))=cos^2(x) is x= pi/2+2pin,x= pi/4+pin

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

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

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

Solution steps

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

Subtract

cos^{2} (x)

from both sides

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

Rewrite using trig identities

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

Factor

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

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

Solving each part separately

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

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

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

Combine all the solutions

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

sin (θ) = \frac{- 1}{2}, 0 \leq θ \leq 4 π

0 = 2 cos (θ)

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

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

y, 2 cos (5 x + y) = 0

What is the general solution for (1-sin(x))(1+cos(x))=cos^2(x) ?
The general solution for (1-sin(x))(1+cos(x))=cos^2(x) is x= pi/2+2pin,x= pi/4+pin