MCQ
$\frac{\sin\theta}{1+\cos\theta}$ is equal to :
- A$\frac{1+\cos\theta}{\sin\theta}$
- B$\frac{1-\cos\theta}{\cos\theta}$
- ✓$\frac{1-\cos\theta}{\sin\theta}$
- D$\frac{1-\sin\theta}{\cos\theta}$
✓
Answer
Correct option: C.
$\frac{1-\cos\theta}{\sin\theta}$
The given expression is $\frac{\sin\theta}{1+\cos\theta}$
Multiplying both the numerator and denominator under the root by $(1-\cos\theta)$, we have
$\frac{\sin\theta}{1+\cos\theta}$
$=\frac{\sin\theta(1-\cos\theta)}{(1+\cos\theta)(1-\cos\theta)}$
$=\frac{\sin\theta(1-\cos\theta)}{1-\cos^2\theta}$
$=\frac{\sin\theta(1-\cos\theta)}{\sin^2\theta}$
$=\frac{1-\cos\theta}{\sin\theta}$
Therefore, the correct option is $(C)$.
Multiplying both the numerator and denominator under the root by $(1-\cos\theta)$, we have
$\frac{\sin\theta}{1+\cos\theta}$
$=\frac{\sin\theta(1-\cos\theta)}{(1+\cos\theta)(1-\cos\theta)}$
$=\frac{\sin\theta(1-\cos\theta)}{1-\cos^2\theta}$
$=\frac{\sin\theta(1-\cos\theta)}{\sin^2\theta}$
$=\frac{1-\cos\theta}{\sin\theta}$
Therefore, the correct option is $(C)$.
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