Prove the following trigonometric identities:$ (\sec \theta+\cos \theta)(\sec \theta-\cos \theta)=\tan ^{2} \theta+\sin ^{2} \theta $

To do:

We have to prove that \( (\sec \theta+\cos \theta)(\sec \theta-\cos \theta)=\tan ^{2} \theta+\sin ^{2} \theta \).

Solution:

We know that,

$\sec^2 \theta-\tan^2 \theta=1$........(i)

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

Therefore,

$(\sec \theta+\cos \theta)(\sec \theta-\cos \theta)=\sec ^{2} \theta-\cos^2 \theta$  ($(a+b)(a-b)=a^2-b^2$)

$=(1+\tan^2 \theta)-(1-\sin^2 \theta)$     (From (i) and (ii))

$=1-1+\tan^2 \theta+\sin^2 \theta$    

$=\tan^2 \theta+\sin^2 \theta$       

Hence proved.      

Tutorialspoint

Updated on: 10-Oct-2022

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