Prove the following. ^3 -1 -1 = ^2 + - Vidyadip

Question

Prove the following.
$\frac{\tan ^3 \theta-1}{\tan \theta-1}=\sec ^2 \theta+\tan \theta$

✓

Answer

Taking LHS
$\frac{\tan ^3 \theta-1}{\tan \theta-1}$
$=\frac{(\tan \theta-1)\left(\tan ^2 \theta+\tan \theta+1\right)}{\tan \theta-1}\left[a^3-b^3=(a-b)\left(a^2+a b+b^2\right)\right]$
$=\tan ^2 \theta+\tan \theta+1$
$=\sec ^2 \theta+\tan \theta\left[1+\tan ^2 \theta=\sec ^2 \theta\right]$
$=\text { RHS }$
Proved.

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