How do you evaluate $ y = \sin (x + (\dfrac{\pi }{2})) $ ?

Hint: Trigonometric functions are used to study the relation between the sides of a right-angled triangle that is the base, the perpendicular and the hypotenuse, there are six trigonometric functions namely sine, cosine, tangent, cosecant, secant and cotangent. Sine, cosine and tangent functions are the main functions while the cosecant, secant and cotangent functions are the reciprocal of these main functions respectively. All the trigonometric functions are interrelated with each other through some identities and to find the trigonometric value of large angles, there are many ways. In the given question, we have to find the sine of the sum of x and $ \dfrac{\pi }{2} $ so we use the related identity to get the correct answer.

Complete step-by-step answer:
Given,
 $ y = \sin (x + (\dfrac{\pi }{2})) $
We know that the sine of the sum of any two angles A and B is equal to the sum of the product of sine of A and cosine of B and the product of cosine of A and sine of B, that is $ \sin (A + B) = \sin A\cos B + \cos A\sin B $
So,
 $
  y = \sin (x + (\dfrac{\pi }{2})) \\
   \Rightarrow y = \sin x\cos \dfrac{\pi }{2} + \cos x\sin \dfrac{\pi }{2} \;
  $
We know that value of $ \cos \dfrac{\pi }{2} = 0 $ and $ \sin \dfrac{\pi }{2} = 1 $ , so –
 $
  y = \sin x(0) + \cos x(1) \\
   \Rightarrow y = \cos x \;
  $
So, the correct answer is “cosx”.

Note: The trigonometric functions are used in many branches of mathematics. In the trigonometric functions, we put different values of angles as x and get a numerical value, to do the inverse process we have inverse trigonometric functions. Different angles give different numerical values or the same angle gives different numerical values for different trigonometric functions. We know the value of the trigonometric functions of some basic angles $ 0,\dfrac{\pi }{6},\dfrac{\pi }{4},\dfrac{\pi }{3}\,and\,\dfrac{\pi }{2} $ . Using this information, we can solve similar questions.