What is the general solution for sin^2(x)-csc^2(x)=tan^2(x)-cot^2(x) ?

The general solution for sin^2(x)-csc^2(x)=tan^2(x)-cot^2(x) is No Solution for x\in\mathbb{R}

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sin^2(x)-csc^2(x)=tan^2(x)-cot^2(x)

sin^{2} (x) - csc^{2} (x) = tan^{2} (x) - cot^{2} (x)

No Solution for x \in R

Solution steps

sin^{2} (x) - csc^{2} (x) = tan^{2} (x) - cot^{2} (x)

Subtract

tan^{2} (x) - cot^{2} (x)

from both sides

sin^{2} (x) - csc^{2} (x) - tan^{2} (x) + cot^{2} (x) = 0

Express with sin, cos

\frac{- sin ^{4} ( x ) + ( - 1 + cos ^{2} ( x ) ) cos ^{2} ( x ) + cos ^{2} ( x ) sin ^{4} ( x )}{cos ^{2} ( x ) sin ^{2} ( x )} = 0

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

- sin^{4} (x) + (- 1 + cos^{2} (x)) cos^{2} (x) + cos^{2} (x) sin^{4} (x) = 0

Factor

- sin^{4} (x) + (- 1 + cos^{2} (x)) cos^{2} (x) + cos^{2} (x) sin^{4} (x) : (- 1 + cos (x)) (1 + cos (x)) (sin^{4} (x) + cos^{2} (x))

(- 1 + cos (x)) (1 + cos (x)) (sin^{4} (x) + cos^{2} (x)) = 0

Solving each part separately

- 1 + cos (x) = 0 or 1 + cos (x) = 0 or sin^{4} (x) + cos^{2} (x) = 0

- 1 + cos (x) = 0 : x = 2 πn

1 + cos (x) = 0 : x = π + 2 πn

sin^{4} (x) + cos^{2} (x) = 0 :

No Solution

Combine all the solutions

x = 2 πn, x = π + 2 πn

Since the equation is undefined for:

2 πn, π + 2 πn

No Solution for x \in R

sin (x) = \frac{53}{54}

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

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

u, x = 4 sin (u)

2 cos (θ) = 2, 0 \leq θ < 2 π

What is the general solution for sin^2(x)-csc^2(x)=tan^2(x)-cot^2(x) ?
The general solution for sin^2(x)-csc^2(x)=tan^2(x)-cot^2(x) is No Solution for x\in\mathbb{R}