Question

Simplify the trigonometric expression $(1 + {\tan ^2}\theta )(1 - \sin \theta )(1 + \sin \theta )$.

Answer 1

Hint: According to the question we have to simplify the trigonometric expression $(1 + {\tan ^2}\theta )(1 - \sin \theta )(1 + \sin \theta )$. So, first of all we have to multiply the terms as given in the trigonometric expression.
Now, to solve the obtained trigonometric expression we have to use the formulas as mentioned below:
Formula used:
$ \Rightarrow (1 + {\tan ^2}\theta ) = {\sec ^2}\theta ...............(A) \\
   \Rightarrow (1 - {\sin ^2}\theta ) = {\cos ^2}\theta ................(B) \\ $
Now, to eliminate ${\sec ^2}\theta $ we have to use the formula as mentioned below:
Formula used:
$ \Rightarrow {\cos ^2}\theta = \dfrac{1}{{{{\sec }^2}\theta }}.........................(C)$

Complete step-by-step solution:
Step 1: First of all we have to multiply the terms which are $(1 - \sin \theta )$ and $(1 + \sin \theta )$ of the given expression as mentioned in the solution hint. Hence,
$ = (1 + {\tan ^2}\theta )({1^2} - {\sin ^2}\theta )..............(1)$
Step 2: Now, to solve the expression (1) as obtained in the step 1 we have to use the formula (A) as mentioned in the solution hint. Hence,
$ = (1 + {\tan ^2}\theta )({\cos ^2}\theta )..............(2)$
Step 3: Now, to solve the expression (2) as obtained in the step 2 we have to use the formula (B) as mentioned in the solution hint. Hence,
$ = ({\sec ^2}\theta )({\cos ^2}\theta )..............(3)$
Step 4: Now, to solve the expression (3) as obtained in the step 3 we have to use the formula (C) as mentioned in the solution hint. Hence,
$ = \dfrac{{({{\sec }^2}\theta )}}{{({{\sec }^2}\theta )}} \\
   = 1 \\ $
Hence, with the help of the formula (A), (B), and (C) we have obtained the value of the given trigonometric expression which is $(1 + {\tan ^2}\theta )(1 - \sin \theta )(1 + \sin \theta ) = 1$.

Note:
 1). To solve the given trigonometric expression it is necessary to substitute the value $(1 + {\tan ^2}\theta ) = {\sec ^2}\theta $ and $(1 - {\sin ^2}\theta ) = {\cos ^2}\theta $ so that we can easily simplify the expression obtained during simplification.
2). It is necessary that we have to simplify the trigonometric terms which are $(1 + {\tan ^2}\theta )$and $(1 - {\sin ^2}\theta )$ which can be converted into ${\sec ^2}\theta $ and ${\cos ^2}\theta $.