What is the general solution for (sec^2(A)+cot^2(A))/(csc^2(A))=sec^2(A) ?

The general solution for (sec^2(A)+cot^2(A))/(csc^2(A))=sec^2(A) is No Solution for A\in\mathbb{R}

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(sec^2(A)+cot^2(A))/(csc^2(A))=sec^2(A)

\frac{sec ^{2} ( A ) + cot ^{2} ( A )}{csc ^{2} ( A )} = sec^{2} (A)

No Solution for A \in R

Solution steps

\frac{sec ^{2} ( A ) + cot ^{2} ( A )}{csc ^{2} ( A )} = sec^{2} (A)

Subtract

sec^{2} (A)

from both sides

\frac{sec ^{2} ( A ) + cot ^{2} ( A )}{csc ^{2} ( A )} - sec^{2} (A) = 0

Simplify

\frac{sec ^{2} ( A ) + cot ^{2} ( A )}{csc ^{2} ( A )} - sec^{2} (A) : \frac{sec ^{2} ( A ) + cot ^{2} ( A ) - sec ^{2} ( A ) csc ^{2} ( A )}{csc ^{2} ( A )}

\frac{sec ^{2} ( A ) + cot ^{2} ( A ) - sec ^{2} ( A ) csc ^{2} ( A )}{csc ^{2} ( A )} = 0

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

sec^{2} (A) + cot^{2} (A) - sec^{2} (A) csc^{2} (A) = 0

Rewrite using trig identities

cot^{2} (A) - cot^{2} (A) sec^{2} (A) = 0

Factor

cot^{2} (A) - cot^{2} (A) sec^{2} (A) : - cot^{2} (A) (sec (A) + 1) (sec (A) - 1)

- cot^{2} (A) (sec (A) + 1) (sec (A) - 1) = 0

Solving each part separately

cot^{2} (A) = 0 or sec (A) + 1 = 0 or sec (A) - 1 = 0

cot^{2} (A) = 0 : A = \frac{π}{2} + πn

sec (A) + 1 = 0 : A = π + 2 πn

sec (A) - 1 = 0 : A = 2 πn

Combine all the solutions

A = \frac{π}{2} + πn, A = π + 2 πn, A = 2 πn

Since the equation is undefined for:

\frac{π}{2} + πn, π + 2 πn, 2 πn

No Solution for A \in R

cos^{3} (x) = - 1

- sin (x) + 3 cos (x) = 0

tan (b) = \frac{5}{12}

sin (θ) = \frac{35}{37}

sin^{2} (θ) = \frac{2 + 3}{4}

What is the general solution for (sec^2(A)+cot^2(A))/(csc^2(A))=sec^2(A) ?
The general solution for (sec^2(A)+cot^2(A))/(csc^2(A))=sec^2(A) is No Solution for A\in\mathbb{R}