Question

The value of \[\sec {45^\circ } \times \sin {45^\circ }\] is equal to:
a) \[1\]
b) \[0\]
c) \[ - 1\]
d) \[\dfrac{1}{{\sqrt 2 }}\]

Answer 1

Hint: Here in such kinds of questions, we are given a trigonometric function whose value needs to be found out. For that we must know the values of various trigonometric functions like sine, cosine, tangent, cotangent, secant, cosecant for various important angles by heart. In this question we just have to put the values of \[\sec 45^\circ \] and \[\sin 45^\circ \] and multiply them to get the desired result.

Complete step-by-step answer:
To solve the given trigonometric function, we are going to directly put the values of \[\sec 45^\circ \] and \[\sin 45^\circ \] in the given trigonometric function. We know that the value of \[\sec 45^\circ \] is \[\sqrt 2 \] and the value of \[\sin 45^\circ \] is \[\dfrac{1}{{\sqrt 2 }}\] . Putting these values in trigonometric function \[\sec {45^\circ } \times \sin {45^\circ }\] ,
we get, \[
  \sec {45^\circ } \times \sin {45^\circ } = \sqrt 2 \times \dfrac{1}{{\sqrt 2 }} \\
   \Rightarrow \sec {45^\circ } \times \sin {45^\circ } = 1 \\
 \]
Thus the value of the given trigonometric function \[\sec {45^\circ } \times \sin {45^\circ }\] comes out to be \[1\] .

So, the correct answer is “Option A”.

Additional information: Trigonometry is a branch of mathematics which we study to learn to deal with relations involving lengths and angles of triangles. It can, in a simpler manner, be called the study of triangles. Various trigonometric functions like sine, cosine, tangent, cotangent, secant, cosecant are defined based on the ratio of sides of right angled triangles.

Note: This is to note that remembering the values of various trigonometric functions like sine, cosine, tangent, cotangent, secant, cosecant for various important angles like \[30^\circ ,45^\circ ,60^\circ ,90^\circ ,0^\circ \] is important. This helps in solving and converting various other complex trigonometric functions.