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
$\text{L.H.S}=\frac{\cos\theta}{1+\sin\theta}$
$=\frac{\cos\theta}{(1+\sin\theta)}\times\frac{1-\sin\theta}{1-\sin\theta}$
$=\frac{(1-\sin\theta)\cos\theta}{1-\sin^2\theta}$
$=\frac{(1-\sin\theta)\cos\theta}{\cos^2\theta}$
$=\frac{(1-\sin\theta)}{\cos\theta}$
$=\text{R.H.S}$
$\therefore\ \text{L.H.S}=\text{R.H.S}$
$=\frac{\cos\theta}{(1+\sin\theta)}\times\frac{1-\sin\theta}{1-\sin\theta}$
$=\frac{(1-\sin\theta)\cos\theta}{1-\sin^2\theta}$
$=\frac{(1-\sin\theta)\cos\theta}{\cos^2\theta}$
$=\frac{(1-\sin\theta)}{\cos\theta}$
$=\text{R.H.S}$
$\therefore\ \text{L.H.S}=\text{R.H.S}$
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