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Question: The value of \(\cos A - \sin A\) when \(A = \dfrac{{5\pi }}{4}\), is: (A) \(\sqrt 2 \) (B) \(\d...

The value of cos⁡A−sin⁡A\cos A - \sin A when A=5π4A = \dfrac{{5\pi }}{4}, is:
(A) 2\sqrt 2
(B) 12\dfrac{1}{{\sqrt 2 }}
(C) 00
(D) 11

Explanation

Solution

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∘{30^ \circ }, 45∘{45^ \circ } and 60∘{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\cos A - \sin A.
So, cos⁡A−sin⁡A\cos A - \sin A
Now, we are also given the value of the angle A as (5π4)\left( {\dfrac{{5\pi }}{4}} \right) radians. So, substituting the value of angle A, we get,
⇒cos⁡5π4−sin⁡5π4\Rightarrow \cos \dfrac{{5\pi }}{4} - \sin \dfrac{{5\pi }}{4}
Now, we know that cos⁡(π+θ)=−cos⁡θ\cos \left( {\pi + \theta } \right) = - \cos \theta . So, using this trigonometric formula in the expression, we get,
⇒cos⁡(π+π4)−sin⁡5π4\Rightarrow \cos \left( {\pi + \dfrac{\pi }{4}} \right) - \sin \dfrac{{5\pi }}{4}
⇒−cos⁡(π4)−sin⁡(π+π4)\Rightarrow - \cos \left( {\dfrac{\pi }{4}} \right) - \sin \left( {\pi + \dfrac{\pi }{4}} \right)
Now, we also know that sin⁡(π+θ)=−sin⁡θ\sin \left( {\pi + \theta } \right) = - \sin \theta . So, we get,
⇒−cos⁡(π4)−[−sin⁡(π4)]\Rightarrow - \cos \left( {\dfrac{\pi }{4}} \right) - \left[ { - \sin \left( {\dfrac{\pi }{4}} \right)} \right]
On opening bracket and simplifying,
⇒−cos⁡(π4)+sin⁡(π4)\Rightarrow - \cos \left( {\dfrac{\pi }{4}} \right) + \sin \left( {\dfrac{\pi }{4}} \right)
Now, we know that the value of cos⁡(π4)\cos \left( {\dfrac{\pi }{4}} \right) is (12)\left( {\dfrac{1}{{\sqrt 2 }}} \right) and value of sin⁡(π4)\sin \left( {\dfrac{\pi }{4}} \right) is (12)\left( {\dfrac{1}{{\sqrt 2 }}} \right). So, we get,
⇒−12+12\Rightarrow - \dfrac{1}{{\sqrt 2 }} + \dfrac{1}{{\sqrt 2 }}
Cancelling the like terms with opposite signs, we get,
⇒0\Rightarrow 0
Hence, the value of cos⁡A−sin⁡A\cos A - \sin A when A=5π4A = \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⁡(π+θ)=−sin⁡θ\sin \left( {\pi + \theta } \right) = - \sin \theta and cos⁡(π+θ)=−cos⁡θ\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.