Prove that + -1 - +1 = 1+ - Vidyadip

Question

Prove that $\frac{\tan\text{A}+\sec\text{A}-1}{\tan\text{A}-\sec\text{A}+1}=\frac{1+\sin\text{A}}{\cos\text{A}}$

✓

Answer

$\text{L.H.S.}=\frac{\tan\text{A}+\sec\text{A}-1}{\tan\text{A}-\sec\text{A}+1}$
$=\frac{\tan\text{A}+\sec\text{A}-(\sec^2\text{A}-\tan^2\text{A})}{(\tan\text{A}-\sec\text{A}+1)}$ $[\because\sec^2\text{A}-\tan^2\text{A}=1]$
$=\frac{(\tan\text{A}+\sec\text{A})-(\sec\text{A}+\tan\text{A})(\sec\text{A}-\tan\text{A})}{(1-\sec\text{A}+\tan\text{A})}$
$=\frac{(\sec\text{A}+\tan\text{A})(1-\sec\text{A}+\tan\text{A})}{1-\sec\text{A}+\tan\text{A}}$
$=\sec\text{A}+\tan\text{A}=\frac{1}{\cos\text{A}}+\frac{\sin\text{A}}{\cos\text{A}}$
$=\frac{1+\sin\text{A}}{\cos\text{A}}=\text{R.H.S.}$

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