Question

Evaluate the value of given expression: $\cos 15^\circ - \sqrt 3 \sin 15^\circ $

Answer 1

Hint: We will start it by writing the given expression and with the help of a trigonometry table for angles and trigonometry difference identity of sine i.e. $\sin \left( {a - b} \right) = \sin a\cos b - \cos a\sin b$, we will be able to evaluate the expression.

Complete step-by-step answer:
Here, we have $\cos 15^\circ - \sqrt 3 \sin 15^\circ $
In order to solve this, first we will take ‘2’ as common from the whole expression, and to balance the expression we will divide the terms of the expression by ‘2’.
$ \Rightarrow 2\left( {\dfrac{1}{2}\cos 15^\circ - \dfrac{{\sqrt 3 }}{2}\sin 15^\circ } \right)$ … (1)
Now, we will substitute the values of fraction in the form of sine and cosine with the help of the trigonometry table for angles in equation (1) i.e. we will put $\dfrac{1}{2} = sin30^\circ $ and $\dfrac{{\sqrt 3 }}{2} = \cos 30^\circ $
$ \Rightarrow 2\left( {sin 30^\circ \cos 15^\circ - \cos 30^\circ \sin 15^\circ } \right)$
As you know, that this expression can be seen as a trigonometry difference identity of sine i.e. $\sin \left( {a - b} \right) = \sin a\cos b - \cos a\sin b$
After, comparing this identity with the expression. So, we can say that
$ \Rightarrow 2\sin \left( {30^\circ - 15^\circ } \right)$, where $a = 30^\circ $ and $b = 15^\circ $
Hence, after simplifying it we can write the given expression as
$ \Rightarrow 2\sin 15^\circ $
Since, after evaluating this given expression we will get $2\sin 15^\circ $.
Hence, we can say that our answer is $2\sin 15^\circ $

Note: We also know that, $\dfrac{1}{2} = \cos 60^\circ $ and $\dfrac{{\sqrt 3 }}{2} = sin60^\circ $
Again by substituting these values in equation (1), we get
$ \Rightarrow 2\left( {\cos 60^\circ \cos 15^\circ - \sin 60^\circ \sin 15^\circ } \right)$
Now we will use the cosine trigonometry sum identity i.e. $\cos \left( {a + b} \right) = \cos a\cos b - \sin a\sin b$ which will leads us to
$ \Rightarrow 2\cos \left( {60^\circ + 15^\circ } \right)$, where $a = 60^\circ $ and $b = 15^\circ $
Therefore, we will get
$ \Rightarrow 2\cos 75^\circ $
Since, after evaluating this given expression we will get $2\cos 75^\circ $.
Hence, we can say that our answer is $2\cos 75^\circ $.
 And we know that $\sin \left( {90^\circ - x} \right) = \cos x$
So, in order to check our answer, we will use the above formula
Since, $sin 15^\circ = \sin \left( {90^\circ - 75^\circ } \right) = \cos 75^\circ $
Hence, our answers are verified.