Question

Find the value of the following expression:
 $\cos 15{}^{\circ }\cos 7{\displaystyle \frac{1{}^{\circ }}{2}}\sin 7{\displaystyle \frac{1{}^{\circ }}{2}}$

Answer 1

Hint: At first try to eliminate the product of $\sin {\displaystyle \frac{15{}^{\circ }}{2}}\cos {\displaystyle \frac{15{}^{\circ }}{2}}$by using the identity $\sin 2\theta =2\sin \theta \cos \theta$ and further use the same identity to convert the expression in terms of $\sin 30{}^{\circ }$ and then use the value of standard value to get the result.

Complete step-by-step solution:
In the question we are asked to find the value of given expression $\cos 15{}^{\circ }\cos 7{\displaystyle \frac{1{}^{\circ }}{2}}\sin 7{\displaystyle \frac{1{}^{\circ }}{2}}$.
To find the value of the expression we need to use the identity $\sin 2\theta =2\sin \theta \cos \theta$.
Now, let’s take the expression,
$\cos 15{}^{\circ }\cos 7{\displaystyle \frac{1{}^{\circ }}{2}}\sin 7{\displaystyle \frac{1{}^{\circ }}{2}}$
$\cos 15{}^{\circ }\cos {\displaystyle \frac{15{}^{\circ }}{2}}\sin {\displaystyle \frac{15{}^{\circ }}{2}}\dots \dots (1)$
Now the expression (1) can be multiplied and divided by 2 so that it’s value remains unaltered so we get,
${\displaystyle \frac{1}{2}}\times 2\times \cos 15{}^{\circ }\cos {\displaystyle \frac{15{}^{\circ }}{2}}\sin {\displaystyle \frac{15{}^{\circ }}{2}}$
which can further be written as,
${\displaystyle \frac{1}{2}}\times \cos 15{}^{\circ }\times (2\cos {\displaystyle \frac{15{}^{\circ }}{2}}\sin {\displaystyle \frac{15{}^{\circ }}{2}})\dots \dots (2)$
Now in the expression (2) we can use the identity $2\sin \theta \cos \theta =\sin 2\theta$ where $\theta$ can be used as $\left({\displaystyle \frac{15{}^{\circ }}{2}}\right)$ so we get,
$\begin{array}{rl}& {\displaystyle \frac{1}{2}}\times \cos 15{}^{\circ }\times \{\sin (2\times {\displaystyle \frac{15}{2}})\}\\ & ={\displaystyle \frac{1}{2}}\times \cos 15{}^{\circ }\times \sin 15{}^{\circ }\dots \dots (3)\end{array}$
Now the expression (3) can be multiplied and divided by 2 so that it’s value remains unaltered so we get,
${\displaystyle \frac{1}{2}}\times {\displaystyle \frac{1}{2}}\times 2\cos 15{}^{\circ }\sin 15{}^{\circ }$
which can be further written as,
${\displaystyle \frac{1}{4}}\times (2\cos 15{}^{\circ }\sin 15{}^{\circ })\dots \dots (4)$
Now in the expression (4) we can use the identity $2\sin \theta \cos \theta =\sin 2\theta$ where $\theta$ can be used $15{}^{\circ }$
So we get,
$\begin{array}{rl}& {\displaystyle \frac{1}{4}}\times \{\sin (2\times 15{}^{\circ })\}\\ & ={\displaystyle \frac{1}{4}}\times \sin 30{}^{\circ }\end{array}$
Here now we will use the value of $\sin 30{}^{\circ }={\displaystyle \frac{1}{2}}$.
So the value of expression is ${\displaystyle \frac{1}{4}}\times {\displaystyle \frac{1}{2}}={\displaystyle \frac{1}{8}}$.

Note: Students while solving this kind of problem have confusion where to start and how to find the value so they should always try to pair up and use the trigonometric identities to find the value. Students should also know the trigonometric identities and formulas by heart. They should also use them wisely to do the problems quickly and easily.