Question

Find the value of $\sin \theta \cos (90 - \theta ) + \cos \theta \sin (90 - \theta )$
A) $0$
B) $1$
C) $2$
D) $\dfrac{3}{2}$

Answer 1

Hint: According to the question we have to determine the value of $\sin \theta \cos (90 - \theta ) + \cos \theta \sin (90 - \theta )$. So, first of all to solve the trigonometric expression we have to use the formula as mentioned below:

Formula used: $
   \Rightarrow \cos ({90^0} - \theta ) = \sin \theta ................(A) \\
   \Rightarrow \sin ({90^0} - \theta ) = \cos \theta ................(B)
 $
Hence, with the help of the formula above we can get the simplified form of the given trigonometric expression now, we have to use the formula as mentioned below to solve the expression.
$ \Rightarrow {\sin ^2}\theta + {\cos ^2}\theta = 1..................(C)$

Complete step-by-step solution:
Step 1: First of all we have to use the formula (A) and (B) as mentioned in the solution hint to simplify the given expression. Hence,
$ = \sin \theta .\sin \theta + \cos \theta .\cos \theta $
Step 2: Now, we have to multiply each term of the expression as obtained in the solution step 2.
$ = {\sin ^2}\theta + {\cos ^2}\theta $
Step 3: Now, to simplify the trigonometric expression as obtained in the solution step 2 we have to use the formula (C) as mentioned in the solution hint.
$ \Rightarrow {\sin ^2}\theta + {\cos ^2}\theta = 1$
Final solution: Hence, with the help of formula (A), (B), and (C) we have obtained the value of the expression $\sin \theta \cos (90 - \theta ) + \cos \theta \sin (90 - \theta )$= 1.

Therefore option (B) is correct.

Note: To simplify the given trigonometric expression it is necessary to convert $\cos (90 - \theta )$ into $\sin \theta $ and $\sin (90 - \theta )$ into $\cos \theta $.
If the given angle is negative for $\cos $ such as $\cos ( - \theta )$ then the angle or value will be always positive as $\cos ( - \theta ) = \cos \theta $