Question

Evaluate $\cos 18^\circ \sin 72^\circ + \sin 18^\circ \cos 72^\circ $

Answer 1

Hint: Here this question can be solved using two methods. First method is by using the concept of complementary angles. Since sine and cosine are complementary, we can write
$ \Rightarrow \sin \left( {90 - \theta } \right) = \cos \theta $ and
$ \Rightarrow \cos \left( {90 - \theta } \right) = \sin \theta $
Other method is by using the direct formula of $\sin \left( {A + B} \right)$. The formula for this is given by
$ \Rightarrow \sin \left( {A + B} \right) = \sin A\cos B + \cos A\cos B$
Both these methods are explained in detail below.

Complete step-by-step solution:
In this question, we are given a trigonometric expression and we need to find its value.
The given expression is: $\cos 18^\circ \sin 72^\circ + \sin 18^\circ \cos 72^\circ $ - - - - - - - - - - - - - - (1)
Now, we do not have the direct values or formulas for this expression. So, we need to use trigonometric formulas and relations to solve this expression.
Here, sine and cosine are complementary to each other. So, therefore, we can take
$ \Rightarrow \sin \left( {90 - \theta } \right) = \cos \theta $ and
$ \Rightarrow \cos \left( {90 - \theta } \right) = \sin \theta $
So, in equation (1), we can adjust the above formula.
In equation (1), we can write $\cos 18 = \sin \left( {90 - 72} \right)$ and $\sin 18 = \cos \left( {90 - 72} \right)$. Therefore equation (1) becomes
$ \Rightarrow \cos 18^\circ \sin 72^\circ + \sin 18^\circ \cos 72^\circ = \sin 72\cos \left( {90 - 72} \right) + \cos 72\sin \left( {90 - 72} \right)$ - - - - - - - - - - - - (2)
Hence, now we can use the above mentioned formulas.
Therefore, $\cos \left( {90 - 72} \right) = \sin 72$ and $\sin \left( {90 - 72} \right) = \cos 72$. Therefore, substituting these values in equation (2), we get
$ \Rightarrow \cos 18^\circ \sin 72^\circ + \sin 18^\circ \cos 72^\circ = \sin 72\sin 72 + \cos 72\cos 72$
$ \Rightarrow \cos 18^\circ \sin 72^\circ + \sin 18^\circ \cos 72^\circ = {\sin ^2}72 + {\cos ^2}72$
Now, we know the identity that
${\sin ^2}\theta + {\cos ^2}\theta = 1$
Therefore, the above equation becomes
$ \Rightarrow \cos 18^\circ \sin 72^\circ + \sin 18^\circ \cos 72^\circ = {\sin ^2}72 + {\cos ^2}72 = 1$

Hence, the value of $\cos 18^\circ \sin 72^\circ + \sin 18^\circ \cos 72^\circ $ is equal to 1.


Note: Here, if this concept seems tricky to you, there is a simple formula too to solve this question.
As we know the formula that $\sin \left( {A + B} \right) = \sin A\cos B + \cos A\cos B$, we can solve this question with this formula.
Here, A=72 and B=18. Therefore,
$ \Rightarrow \cos 18^\circ \sin 72^\circ + \sin 18^\circ \cos 72^\circ = \sin \left( {72 + 18} \right)$
$
   = \sin 90 \\
   = 1 \\
 $
Hence, $\cos 18^\circ \sin 72^\circ + \sin 18^\circ \cos 72^\circ $ is equal to 1.