Prove the following trigonometric identities. (1+ ^2 ) cosec^2 = - Vidyadip

Question

Prove the following trigonometric identities.
$\frac{(1+\tan^2\theta)\cot\theta}{\text{cosec}^2\theta}=\tan\theta$

✓

Answer

We need to prove $\frac{(1+\tan^2\theta)\cot\theta}{\text{cosec}^2\theta}=\tan\theta$
Solving the L.H.S, we get
$\text{L.H.S}=\frac{(1+\tan^2\theta)\cot\theta}{\text{cosec}^2\theta}=\frac{\sec^2\theta(\cot\theta)}{\text{cosec}^2\theta}$
Using $\sec\theta=\frac{1}{\cos\theta},\cot\theta=\frac{\cos\theta}{\sin\theta}\text{ and } \text{cosec}\theta=\frac{1}{\sin\theta}$, we get
$\frac{\sec^2\theta(\cot\theta)}{\text{cosec}^2\theta}=\frac{\frac{1}{\cos^2\theta}\Big(\frac{\cos\theta}{\sin\theta}\Big)}{\frac{1}{\sin^2\theta}}$
$=\frac{\frac{1}{\cos\theta\sin\theta}}{\frac{1}{\sin^2\theta}}$
$=\frac{\sin^2\theta}{\cos\theta\sin\theta}$
$=\frac{\sin\theta}{\cos\theta}$
$=\tan\theta=\text{R.H.S}$
$\therefore\ \text{L.H.S.}=\text{R.H.S}$

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