Prove the following identities. ^4 + ^2 = ^4 - ^2 - Vidyadip

Question

Prove the following identities.
$\tan ^4 \theta+\tan ^2 \theta=\sec ^4 \theta-\sec ^2 \theta$

✓

Answer

$\tan ^4 \theta+\tan ^2 \theta=\sec ^4 \theta-\sec ^2 \theta$
$\text { L.H.S }=\tan ^4 \theta+\tan ^2 \theta$
Taking out $\tan ^2 \theta$ as common
$=\tan ^2 \theta\left(\tan ^2 \theta+1\right)$
We know that
$1+\tan ^2 \theta=\sec ^2 \theta$
$\text { i.e. } \tan ^2 \theta=\sec ^2 \theta-1$
It can be written as
$=\left(\sec ^2 \theta-1\right) \sec ^2 \theta$
So we get
$=\sec ^4 \theta-\sec ^2 \theta$
$=\text { R.H.S }$
Therefore, it is proved.

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