Question

How do you simplify $\cos \left( {x - \dfrac{\pi }{2}} \right)$?

Answer 1

Hint: Here we can proceed by applying the formula $\cos (A - B) = \cos A\cos B + \sin A\sin B$
And then we can easily substitute the value of $\cos \dfrac{\pi }{2} = 0$ and $\sin \dfrac{\pi }{2} = 1$ and get our desired result.

Complete step-by-step answer:
Here we are given to simplify the value of the term $\cos \left( {x - \dfrac{\pi }{2}} \right)$
We know that there are different formulas of all the trigonometric functions and we need to just know which formula must be applied and when to apply which formula. So as we are given the term $\cos \left( {x - \dfrac{\pi }{2}} \right)$ which is of the form $\cos (A - B)$ so we can apply the formula of this trigonometric function which is:
$\cos (A - B) = \cos A\cos B + \sin A\sin B$$ - - - - (1)$
So by comparing this $\cos (A - B)$ with the term $\cos \left( {x - \dfrac{\pi }{2}} \right)$ we can get the values of $A{\text{ and }}B$ as:
$
  A = x \\
  B = \dfrac{\pi }{2} \\
 $
Now substituting the values of them in equation (1) we will get:
$\cos (x - \dfrac{\pi }{2}) = \cos x\cos \dfrac{\pi }{2} + \sin x\sin \dfrac{\pi }{2}$$ - - - - (2)$
We also know that $\cos \dfrac{\pi }{2} = 0$ and $\sin \dfrac{\pi }{2} = 1$
Now substituting these values in equation (2) we will get:
$\cos (x - \dfrac{\pi }{2}) = \cos x(0) + \sin x(1)$
Simplifying it we get:
$\cos (x - \dfrac{\pi }{2}) = \sin x$
Hence on simplifying we have got the value of $\cos (x - \dfrac{\pi }{2}) = \sin x$ and therefore we must remember that in order to solve such problems that deals with the trigonometric functions we need to always remember the trigonometric formula as without them, the problem is quite difficult to solve. We also have another formula in $\sin ,\cos ,\tan ,\cot ,\sec ,{\text{cosec}}$
Moreover when we are asked to find the value of the tangent or cotangent of any angle, we must know the basic values of the sine and cosine of the angles like $0^\circ ,30^\circ ,45^\circ ,60^\circ ,90^\circ $ and then we can easily calculate the same angles of the tangent, cotangent, secant, and cosecant of that same angle.

Hence we get the value of $\cos (x - \dfrac{\pi }{2}) = \sin x$

Note: We can also solve this by another method where we can use two formula:
1) $\cos (\dfrac{\pi }{2} - x) = \sin x$
2) $\cos ( - x) = \cos x$
So we can write this given problem as:
$\cos (x - \dfrac{\pi }{2}) = \cos ( - (\dfrac{\pi }{2} - x)) = \cos (\dfrac{\pi }{2} - x) = \sin x$