Prove the following identities: ^4x− ^2x= ^4x+ ^2x - Vidyadip

Question

Prove the following identities:
$\sec^4\text{x}− \sec^2\text{x}= \tan^4\text{x}+\tan^2\text{x}$

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Answer

$\text{L.H.S.} = \sec^4\text{x} - \sec^{2}\text{x}$
$=\sec^{2}\text{x}\big(\sec^{2}\text{x}-1\big)$
$=\big(1+\tan^{2}\text{x}\big)\tan^{2}\text{x}$ $\big[\because\sec^{2}\text{x}=1+\tan^{2}\text{x}\big]$
$=\tan^{2}\text{x}+\tan^{4}\text{x} $
$=\tan^{4}\text{x}+\tan^{2}\text{x}$
$=\text{R.H.S}$
$\text{L.H.S}=\text{R.H.S}$

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