Question

# The value of $\left( {\sec A + \tan A} \right)\left( {1 - \sin A} \right)$ is equal to:A) Sec AB) Sin AC) Cosec AD) Cos A

Hint: In the given problem we will use trigonometric formulas and put them in the given equation and solve it. Since there are a number of formulas in trigonometry, so we can also apply the formula according to the given options and solve the equation. Thus we will get the correct answer.

Complete step by step solution: Given equation:
$\left( {\sec A + \tan A} \right)\left( {1 - \sin A} \right)$
We know that:
$\sec A = \dfrac{1}{{\cos A}} \\ and \\ \tan A = \dfrac{{\operatorname{sinA} }}{{\cos A}} \\$
Put these values in the given equation.
We get:
$\left( {\dfrac{1}{{\cos A}} + \dfrac{{\sin A}}{{\operatorname{cosA} }}} \right)\left( {1 - \sin A} \right) \\ \Rightarrow \left( {\dfrac{{1 + \sin A}}{{\cos A}}} \right)\left( {1 - \sin A} \right) \\ \Rightarrow \dfrac{{\left( {1 - {{\sin }^2}A} \right)}}{{\cos A}} \\$
As
$\left( {1 - {{\sin }^2}A} \right) = {\cos ^2}A$
We get
$\dfrac{{{{\cos }^2}A}}{{\cos A}} \\ \Rightarrow \cos A \\$

Hence the answer is $\cos A$ and the correct option is D.

Note: First we know that and we have to remember all the trigonometric formulas because sometimes answer is already given in the options; we only apply the appropriate formula for the correct answer. Here we apply the formula according to the options given i.e. the formula for secA and tan A. Hence we get the correct answer.