Question

The value of $\cos {15^ \circ } - \sin {15^ \circ }$ is equal to
A.$\dfrac{1}{{\sqrt 2 }}$
B.$\dfrac{1}{2}$
C.$\dfrac{{ - 1}}{{\sqrt 2 }}$
D.$0$

Answer 1

Here in this question, we have to find the exact value of a given trigonometric function. For this first we have to the angle of cosine function in terms of sum or difference of complementary angle ${90^ \circ }$ and further simplify by using a Sum to Product Formula of trigonometry i.e., $\sin x - \sin y = 2\cos \left( {\dfrac{{x + y}}{2}} \right)\sin \left( {\dfrac{{x - y}}{2}} \right)$ and by the standard angles values of trigonometric ratios we get the required value.

Complete answer:
A function of an angle expressed as the ratio of two of the sides of a right triangle that contains that angle; the sine, cosine, tangent, cotangent, secant, or cosecant known as trigonometric function Also called circular function.
Consider the given question:
$\cos {15^ \circ } - \sin {15^ \circ }$ --------(1)
$\cos {15^ \circ }$ can be written in difference of ${90^ \circ }$ is $\cos {15^ \circ } = \cos \left( {{{90}^ \circ } - {{75}^ \circ }} \right)$, then equation (1) becomes
$\Rightarrow \,\,\,\cos \left( {{{90}^ \circ } - {{75}^ \circ }} \right) - \sin {15^ \circ }$ -----(2)
Let us by the complementary angles of trigonometric ratios:
The angle can be written as
$\sin \left( {90 - \theta } \right) = \cos \theta$
$\cos \left( {90 - \theta } \right) = \sin \theta$
On substituting in equation (2), we have
$\Rightarrow \,\,\,\sin {75^ \circ } - \sin {15^ \circ }$ -----(3)
Now, apply the sum to product formula of trigonometry i.e., $\sin x - \sin y = 2\cos \left( {\dfrac{{x + y}}{2}} \right)\sin \left( {\dfrac{{x - y}}{2}} \right)$
Here, $x = {75^ \circ }$ and $y = {15^ \circ }$
On substituting the $x$ and $y$ values in formula, we have
$\Rightarrow \,\,\,2\cos \left( {\dfrac{{75 + 15}}{2}} \right)\sin \left( {\dfrac{{75 - 15}}{2}} \right)$
$\Rightarrow \,\,\,2\cos \left( {\dfrac{{90}}{2}} \right)\sin \left( {\dfrac{{60}}{2}} \right)$
On simplification, we get
$\Rightarrow \,\,\,2\cos \left( {{{45}^ \circ }} \right)\sin \left( {{{30}^ \circ }} \right)$ -----(4)
As we know, from the standard angles table of trigonometric ratios the value of $\cos {45^ \circ } = \dfrac{1}{{\sqrt 2 }}$ and $\sin {30^ \circ } = \dfrac{1}{2}$.
On substituting the values in equation (4), then
$\Rightarrow \,\,\,2 \cdot \left( {\dfrac{1}{{\sqrt 2 }}} \right) \cdot \left( {\dfrac{1}{2}} \right)$
On simplification, we get
$\Rightarrow \,\,\,\dfrac{1}{{\sqrt 2 }}$
Hence, the value of $\cos {15^ \circ } - \sin {15^ \circ } = \dfrac{1}{{\sqrt 2 }}$.
Therefore, option A is the correct answer.

Note:
When solving the trigonometry-based questions, we have to know the definitions and table of standard angles of all six trigonometric ratios. Remember, when the sum of two angles is ${90^ \circ }$, then the angles are known as complementary angles at that time the ratios will change like $\sin \leftrightarrow \cos$, $\sec \leftrightarrow cosec$ and $\tan \leftrightarrow \cot$ then should know the some basic formulas of trigonometry like identities, double and half angle formulas, Product to Sum Formulas and Sum to Product Formulas.