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

The general solution for cot^2(x)=(1+sin(x))/(1+cos(x)) is x=(3pi)/2+2pin,x= pi/4+pin

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cot^2(x)=(1+sin(x))/(1+cos(x))

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

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

Solution steps

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

Subtract

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

from both sides

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

Simplify

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

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

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

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

Express with sin, cos

\frac{cos ^{2} ( x ) + cos ^{3} ( x ) - sin ^{2} ( x ) - sin ^{3} ( x )}{sin ^{2} ( x )} = 0

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

cos^{2} (x) + cos^{3} (x) - sin^{2} (x) - sin^{3} (x) = 0

Factor

cos^{2} (x) + cos^{3} (x) - sin^{2} (x) - sin^{3} (x) : (- sin (x) + cos (x)) (sin^{2} (x) + sin (x) cos (x) + sin (x) + cos^{2} (x) + cos (x))

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

Rewrite using trig identities

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

Solving each part separately

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

cos (x) + sin (x) + cos (x) sin (x) + 1 = 0 : x = \frac{3 π}{2} + 2 πn, x = π + 2 πn

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

Combine all the solutions

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

Since the equation is undefined for:

π + 2 πn

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

0.7 sin (3 π t) - 3 π cos (3 π t) = 0

sin (θ) = \frac{8}{15}

cos (x) cot (x) cos (x) = 0, 0 \leq x \leq 2 π

0 = cos^{2} (x)

3 cos (θ) = - 3 sin (- θ)

What is the general solution for cot^2(x)=(1+sin(x))/(1+cos(x)) ?
The general solution for cot^2(x)=(1+sin(x))/(1+cos(x)) is x=(3pi)/2+2pin,x= pi/4+pin