Question

Please please 8 this one prove that

Answer 1

Q) $\boxed{\sf{ \frac{1 + tan ^{2}(45\degree - A) }{1 - tan ^{2}(45\degree - A) } = cosec\:2A}}$

$\sf{\underline{LHS:}}$

$\sf{= \frac{1 + tan ^{2}(45\degree - A) }{1 - tan ^{2}(45\degree - A) }}$

$\sf{ = \frac{1}{cos2(45\degree - A)}}$

$\sf{\underline{Since:}}$ $\boxed{\sf{ \frac{1 - {tan}^{2}A }{1 + {tan}^{2} A} = cos\:2A}}$

$\sf{= \frac{1}{cos(90\degree - 2A)}}$

$\sf{= \frac{1}{sin\:2A}}$

$\sf{ = cosec\:2A}$

$\sf{ = RHS}$

Hence, Proved!