Prove that tan A / (1+ sec A) - tan A / (1- sec A) = 2 cosec A

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Q) Prove that $\frac {\tan A} {1+ \sec A} - \frac{\tan A}{1- \sec A}$ = 2 cosec A

Ans: Let’s start from LHS

LHS = $\frac {\tan A} {1+ \sec A} - \frac{\tan A}{1- \sec A}$

= $\frac {\tan A (1 - \sec A) - \tan A (1 + \sec A)} {(1+ \sec A) (1 - \sec A)}$

= $\frac {- 2 \tan A \sec A} {1 - \sec ^2 A}$

= $\frac {- 2 \tan A \sec A} {- \tan ^2 A}$

= $\frac {2 \sec A} {\tan A}$

= $\frac {2} {\sin A}$

= 2 cosec A

= RHS ……….. Hence, proved !

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