Question

The expression $E=\frac{1+\cos \theta +\sin \theta }{1+\cos \theta -\sin \theta }$ can be simplified to

• A. cosθ/(1+sinθ)
• B. (1+cosθ)/sinθ
• C. sinθ/(1+cosθ)
• D. (1+sinθ)/cosθ

Answer 1

The correct option is D

(1+sinθ)/cosθ

We multiply the numerator and denominator of E by cos $\theta$ -

$\begin{array}{c}E=\frac{cos\theta \left(1+cos\theta +sin\theta \right)}{cos\theta \left(1+cos\theta -sin\theta \right)}\\ =\frac{cos\theta +sin\theta .cos\theta +co{s}^2\theta }{cos\theta \left(1+cos\theta -sin\theta \right)}\\ =\frac{cos\theta \left(1+sin\theta \right)+\left(1-si{n}^2\theta \right)}{cos\theta \left(1+cos\theta -sin\theta \right)}\\ =\frac{\left(1+sin\theta \right)\left(cos\theta +1-sin\theta \right)}{cos\theta \left(1+cos\theta -sin\theta \right)}\\ =\frac{1+sin\theta }{cos\theta }\end{array}$