Question

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

• A. cosθ/(1-sinθ)
• B. sinθ/(1-cosθ)
• C. (1-sinθ)/cosθ
• D. (1-cosθ)/sinθ
• E. None of these

Answer 1

The correct option is A

cosθ/(1-sinθ)

We write the 1 in the numerator as (sec${}^2$$\theta$ - tan${}^2$$\theta$) -

$\begin{array}{c}E=\frac{sec\theta +tan\theta -(se{c}^2\theta -ta{n}^2\theta )}{tan\theta -sec\theta +1}\\ =\frac{sec\theta +tan\theta -\left(sec\theta +tan\theta \right)\left(sec\theta -tan\theta \right)}{1-sec\theta +tan\theta }\\ =\frac{\left(sec\theta +tan\theta \right)\left(1-sec\theta +tan\theta \right)}{1-sec\theta +tan\theta }\\ =sec\theta +tan\theta \\ =\frac{1}{sec\theta -tan\theta }\\ =\frac{cos\theta }{1-sin\theta }\end{array}$