Question

sin^2÷coc^2 + cos^2÷sin^2 = sec^2-cosec^2-2​

Answer 1

$\underline{\bold{Solution:}}$

$\text{Now, L.H.S.}$

$\bold{ = \dfrac{sin^{2}\theta}{cos^{2}\theta} + \dfrac{cos^{2}\theta}{sin^{2}\theta}}$

$\bold{= tan^{2}\theta + cot^{2}\theta}$

$\bold{= sec^{2}\theta - 1 + 1 - cosec^{2}\theta}$

$\boxed {\tiny{\bold{Since, \: sec^{2}\theta - tan^{2}\theta = 1 \:\: \& \:\: cosec^{2}\theta - cot^{2}\theta = 1 }}}$

$\bold{= sec^{2}\theta - cosec^{2}\theta}$

$\text{= R.H.S}$

$\to \boxed{\small{\bold{\dfrac{sin^{2}\theta}{cos^{2}\theta} + \dfrac{cos^{2}\theta}{sin^{2}\theta} = sec^{2}\theta - cosec^{2}\theta}}}$

$\bold{\{PROVED\}}$