What is the general solution for sin^3(x)+cos^3(x)=(1-1)/(2sin^2(x)) ?

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

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sin^3(x)+cos^3(x)=(1-1)/(2sin^2(x))

sin^{3} (x) + cos^{3} (x) = \frac{1 - 1}{2 sin ^{2} ( x )}

x = \frac{3 π}{4} + πn

Solution steps

sin^{3} (x) + cos^{3} (x) = \frac{1 - 1}{2 sin ^{2} ( x )}

Subtract

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

from both sides

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

Factor

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

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

Rewrite using trig identities

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

Solving each part separately

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

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

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

No Solution

Combine all the solutions

x = \frac{3 π}{4} + πn

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

c, sin^{2} (x) + 12 = 12 (sin (x) - cos (x))

c, sin (x) - 5 cos (x) = sin^{2} (x) + 5 cos^{2} (x)

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

y = 2 sin (cos (+ 0))

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