Question

The value of $\cos A - \sin A$ when $A = \dfrac{{5\pi }}{4}$, is:
(A) $\sqrt 2 $
(B) $\dfrac{1}{{\sqrt 2 }}$
(C) $0$
(D) $1$

Answer 1

Hint: The given question deals with basic simplification of trigonometric functions by using some of the simple trigonometric formulae and identities. Basic algebraic rules and trigonometric identities are to be kept in mind while doing simplification in the given problem. We must know the values of trigonometric ratios for some standard and basic angles like ${30^ \circ }$, ${45^ \circ }$ and ${60^ \circ }$. So, we will convert the trigonometric functions of the given angle into standard angles and then simplify the value to get to the final answer.

Complete answer:
In the given problem, we have to simplify the value of $\cos A - \sin A$.
So, $\cos A - \sin A$
Now, we are also given the value of the angle A as $\left( {\dfrac{{5\pi }}{4}} \right)$ radians. So, substituting the value of angle A, we get,
$ \Rightarrow \cos \dfrac{{5\pi }}{4} - \sin \dfrac{{5\pi }}{4}$
Now, we know that $\cos \left( {\pi + \theta } \right) = - \cos \theta $. So, using this trigonometric formula in the expression, we get,
$ \Rightarrow \cos \left( {\pi + \dfrac{\pi }{4}} \right) - \sin \dfrac{{5\pi }}{4}$
$ \Rightarrow - \cos \left( {\dfrac{\pi }{4}} \right) - \sin \left( {\pi + \dfrac{\pi }{4}} \right)$
Now, we also know that $\sin \left( {\pi + \theta } \right) = - \sin \theta $. So, we get,
$ \Rightarrow - \cos \left( {\dfrac{\pi }{4}} \right) - \left[ { - \sin \left( {\dfrac{\pi }{4}} \right)} \right]$
On opening bracket and simplifying,
$ \Rightarrow - \cos \left( {\dfrac{\pi }{4}} \right) + \sin \left( {\dfrac{\pi }{4}} \right)$
Now, we know that the value of $\cos \left( {\dfrac{\pi }{4}} \right)$ is $\left( {\dfrac{1}{{\sqrt 2 }}} \right)$ and value of $\sin \left( {\dfrac{\pi }{4}} \right)$ is $\left( {\dfrac{1}{{\sqrt 2 }}} \right)$. So, we get,
$ \Rightarrow - \dfrac{1}{{\sqrt 2 }} + \dfrac{1}{{\sqrt 2 }}$
Cancelling the like terms with opposite signs, we get,
$ \Rightarrow 0$
Hence, the value of $\cos A - \sin A$ when $A = \dfrac{{5\pi }}{4}$ is zero.
Hence, option (D) is the correct answer.

Note:
For solving the given question, we must know the some basic and simple trigonometric formulae such as $\sin \left( {\pi + \theta } \right) = - \sin \theta $ and $\cos \left( {\pi + \theta } \right) = - \cos \theta $. One must also know the values of trigonometric functions for some standard and basic angles to solve such questions. We should take care of the calculations in order to get to the correct answer. We should also know the periodicity of trigonometric functions as it helps in solving this type of questions.