Question

prove that following identities ,Where the angles involved are acute angle for Which the expressions are defined

Answer 1

OK I have bit tried it to solve it bro

Answer 2

The easier one:

$\frac{1 + \tan^{2}(A)}{1 + \cot^{2}(A)}\right = \frac{\sec^2(A)}{\csc^{2}(A)}  = \frac{1}{\cos^{2}(A)} . \sin^{2}(A) = \tan^{2}(A)$,

as required.

The slightly harder one:

$\left(\frac{1 - \tan(A)}{1 - \cot(A)}\right)^{2}$

$= \frac{1 - 2\tan(A) + \tan^{2}(A)}{1 - 2\cot(A) + \cot^{2}(A)}$

$= \frac{\sec^{2}(A) - 2\tan(A)}{\csc^{2}(A) - 2\cot(A)}$

$= \frac{\frac{1}{\cos^{2}(A)} - \frac{2\sin(A)}{\cos(A)}}{\frac{1}{\sin^{2}(A)} - \frac{2\cos(A)}{\sin(A)}}$

$= \frac{1 - 2\sin(A)\cos(A)}{\cos^{2}(A)}.\frac{\sin^{2}(A)}{1 - 2\sin(A)\cos(A)}$

$= \frac{\sin^{2}(A)}{\cos^{2}(A)}$

$= \underline{\underline{\tan^{2}(A)}},$

as required.