Question
Prove the following trigonometric identities.
$(\text{cosec}\text{A}-\sin\text{A})(\sec\text{A}-\cos\text{A})(\tan\text{A}+\cot\text{A})= 1$
$(\text{cosec}\text{A}-\sin\text{A})(\sec\text{A}-\cos\text{A})(\tan\text{A}+\cot\text{A})= 1$
✓
Answer
$\text{L.H.S.}=(\text{cosec A}-\sin\text{A})(\sec\text{A}-\cos\text{A})(\tan\text{A}+\cot\text{A})$
$=\Big[\frac{1}{\sin\text{A}}-\sin\text{A}\Big]\Big[\frac{1}{\cos\text{A}}-\cos\text{A}\Big]\Big[\frac{\sin\text{A}}{\cos\text{A}}+\frac{\cos\text{A}}{\sin\text{A}}\Big]$
$\Big(\frac{1-\sin^2\text{A}}{\sin\text{A}}\Big)\Big(\frac{1-\cos^2\text{A}}{\cos\text{A}}\Big)\Big(\frac{\sin^2\text{A}+\cos^2\text{A}}{\sin\text{A}\cos\text{A}}\Big)$
$\frac{\cos^2\text{A}}{\sin\text{A}}\times\frac{\sin^2\text{A}}{\cos\text{A}}\times\frac{1}{\sin\text{A}\cos\text{A}}$
$\frac{\cos^2\text{A}\cdot\sin^2\text{A}}{\cos^2\text{A}\cdot\sin^2\text{A}}$
$=1=\text{R.H.S}.$
$=\Big[\frac{1}{\sin\text{A}}-\sin\text{A}\Big]\Big[\frac{1}{\cos\text{A}}-\cos\text{A}\Big]\Big[\frac{\sin\text{A}}{\cos\text{A}}+\frac{\cos\text{A}}{\sin\text{A}}\Big]$
$\Big(\frac{1-\sin^2\text{A}}{\sin\text{A}}\Big)\Big(\frac{1-\cos^2\text{A}}{\cos\text{A}}\Big)\Big(\frac{\sin^2\text{A}+\cos^2\text{A}}{\sin\text{A}\cos\text{A}}\Big)$
$\frac{\cos^2\text{A}}{\sin\text{A}}\times\frac{\sin^2\text{A}}{\cos\text{A}}\times\frac{1}{\sin\text{A}\cos\text{A}}$
$\frac{\cos^2\text{A}\cdot\sin^2\text{A}}{\cos^2\text{A}\cdot\sin^2\text{A}}$
$=1=\text{R.H.S}.$
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