Question

Prove that, (1+tan^2 theta)(1+cot^2 theta) = 1/ (sin^2 theta - sin^4 theta)​

Answer 1

Step-by-step explanation:

We have,

$\{1 + \tan ^{2} ( \theta ) \} \{1 + \cot ^{2} ( \theta ) \}$

$= \{ \sec ^{2} ( \theta ) \} \{ \cosec ^{2} ( \theta ) \}$

$= \dfrac{1}{ \cos ^{2} ( \theta )}. \dfrac{1}{\sin ^{2} ( \theta ) }$

$= \dfrac{1}{ \cos ^{2} ( \theta )\sin ^{2} ( \theta ) }$

$= \dfrac{1}{ \{1 - \sin^{2} ( \theta ) \}\sin ^{2} ( \theta ) }$

$= \dfrac{1}{\sin ^{2} ( \theta ) - \sin^{4} ( \theta ) }$