Question
Prove the following trigonometric identities.
$(1-\cos^2\text{A})\text{cosec}^2\text{A}=1$
$(1-\cos^2\text{A})\text{cosec}^2\text{A}=1$
✓
Answer
We know $\sin^2\text{A}+\cos^2\text{A}=1$
$\text{L.H.S.} = (1-\cos^2\text{A})\text{cosec}^2\text{A}$
$\sin^2\text{A}=1-\cos^2\text{A}$
$=\ \sin^2\text{A}.\text{cosec}^2\text{A}$
$=\ \sin^2\text{A}.\frac{1}{\sin^2\text{A}}=1$
$=1= \text{R.H.S.}$
$$$\therefore\ \text{L.H.S.}=\text{R.H.S}$
$\text{L.H.S.} = (1-\cos^2\text{A})\text{cosec}^2\text{A}$
$\sin^2\text{A}=1-\cos^2\text{A}$
$=\ \sin^2\text{A}.\text{cosec}^2\text{A}$
$=\ \sin^2\text{A}.\frac{1}{\sin^2\text{A}}=1$
$=1= \text{R.H.S.}$
$$$\therefore\ \text{L.H.S.}=\text{R.H.S}$
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