Question
Prove the following trigonometric identities.
$\frac{\cos\theta}{1-\sin\theta}=\frac{1+\sin\theta}{\cos\theta}$
$\frac{\cos\theta}{1-\sin\theta}=\frac{1+\sin\theta}{\cos\theta}$
✓
Answer
$\text{L.H.S}=\frac{\cos\theta}{1-\sin\theta}$
Taking rationalisation
$=\frac{\cos\theta(1+\sin\theta)}{(1-\sin\theta)(1+\sin\theta)}$
$=\frac{\cos\theta(1+\sin\theta)}{1-\sin^2\theta}$
$=\frac{\cos\theta(1+\sin\theta)}{\cos^2\theta}$
$=\frac{1+\sin\theta}{\cos\theta}=\text{R.H.S}$
Taking rationalisation
$=\frac{\cos\theta(1+\sin\theta)}{(1-\sin\theta)(1+\sin\theta)}$
$=\frac{\cos\theta(1+\sin\theta)}{1-\sin^2\theta}$
$=\frac{\cos\theta(1+\sin\theta)}{\cos^2\theta}$
$=\frac{1+\sin\theta}{\cos\theta}=\text{R.H.S}$
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