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