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

Hint: 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.