Question

Prove the following: $\sec A\left( 1-\sin A \right)\left( \sec A+\tan A \right)=1$

Answer 1

In this question we have been given with a trigonometric expression which we have to solve and prove. We will solve this question by first considering the left-hand side of the expression and use the various trigonometric identities of sec and tan. We will first multiply the terms and convert the term using the appropriate formula. We will then use the formula $\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$ to get the expression in the appropriate form and get the required solution.

Complete step by step answer:
We have the expression given to us as:
$\Rightarrow \sec A\left( 1-\sin A \right)\left( \sec A+\tan A \right)=1$
Consider the left-hand side of the expression. We get:
$\Rightarrow \sec A\left( 1-\sin A \right)\left( \sec A+\tan A \right)$
On multiplying the first two terms in the expression, we get:
$\Rightarrow \left( \sec A-\sec A\sin A \right)\left( \sec A+\tan A \right)$
Now we know the trigonometric identity that $\sec A=\dfrac{1}{\cos A}$ therefore, on substituting, we get:
$\Rightarrow \left( \sec A-\dfrac{1}{\cos A}\sin A \right)\left( \sec A+\tan A \right)$
On simplifying the terms, we get:
$\Rightarrow \left( \sec A-\dfrac{\sin A}{\cos A} \right)\left( \sec A+\tan A \right)$
Now we know the trigonometric identity that $\tan A=\dfrac{\sin A}{\cos A}$ therefore, on substituting, we get:
$\Rightarrow \left( \sec A-\tan A \right)\left( \sec A+\tan A \right)$
Now we can see that the above expression is in the form of $\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$ therefore, we can write the expression as:
$\Rightarrow {{\sec }^{2}}A-{{\tan }^{2}}A$
Now we know the trigonometric identity that ${{\sec }^{2}}A=1+{{\tan }^{2}}A$ which can be rewritten as ${{\sec }^{2}}A-{{\tan }^{2}}A=1$ therefore, on substituting, we get:
$\Rightarrow 1$, which is the right-hand side of the expression, hence proved.

Note: The various trigonometric identities and formulae should be remembered while doing these types of sums. the various Pythagorean identities should also be remembered while doing these types of questions. To simplify any given equation, it is good practice to convert all the identities into sin and cos for simplifying. If there is nothing to simplify, then only you should use the double angle formulas to expand the given equation.