Prove the following trigonometric identities. 1+ - 1+ + =1- - Vidyadip

Question

Prove the following trigonometric identities.
$\frac{1+\sec\theta-\tan\theta}{1+\sec\theta+\tan\theta}=\frac{1-\sin\theta}{\cos\theta}$

✓

Answer

Cosider the L.H.S
$\text{L.H.S}=\frac{1+\sec\theta-\tan\theta}{1+\sec\theta+\tan\theta}$
$=\frac{(\sec\theta-\tan\theta)+1}{1+\sec\theta+\tan\theta}$
$=\frac{(\sec\theta-\tan\theta)+(\sec^2\theta-\tan^2\theta)}{1+\sec\theta+\tan\theta} \ (\sec^2\theta-\tan^2\theta=1)$
$=\frac{(\sec\theta-\tan\theta)(1+\sec\theta+\tan\theta)}{1+\sec\theta+\tan\theta}$
$=\sec\theta-\tan\theta$
$=\frac{1}{\cos\theta}-\frac{\sin\theta}{\cos\theta}$
$=\frac{1-\sin\theta}{\cos\theta}=\text{R.H.S}$
$\therefore\ \text{L.H.S}=\text{R.H.S}$

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