Answer

Verified

338.4k+ views

**Hint:**To solve the given question, we should know some of the trigonometric properties of sine and cosine function. We should know that \[\sin \left( \dfrac{\pi }{2}-x \right)=\cos x\]. In other words, if the sine of an angle is equal to the cosine of another angle, then the two angles should be complementary. We will use this property to solve this question.

**Complete step-by-step solution:**

We are asked to prove the statement. To prove a statement, we need to show that either the left-hand side or right-hand side of the given statement can be simplified to express the other side. Here the left-hand side and right-hand side of the statement are \[{{\sin }^{-1}}x+{{\cos }^{-1}}x\]and \[\dfrac{\pi }{2}\] respectively. Let’s take the LHS to simplify, assume that \[{{\sin }^{-1}}x=A\], and \[{{\cos }^{-1}}x=B\]. Taking sine of both sides of the first equation, and cosine of both sides of the second equation, we get \[\sin A=x\And \cos B=x\]. Thus, we can say that \[\sin A=\cos B\]. Using the property that if the sine of angle is equal to cosine of another angle, then the two angles should be complementary. We can say that

\[\Rightarrow A+B=\dfrac{\pi }{2}\]

From the substitution, we have \[{{\sin }^{-1}}x=A\], and \[{{\cos }^{-1}}x=B\]. Thus, the above expression can be written as

\[\Rightarrow {{\sin }^{-1}}x+{{\cos }^{-1}}x=\dfrac{\pi }{2}=RHS\]

\[\therefore LHS=RHS\]

Hence proved.

**Note:**The statement given in the above question is one of the properties of inverse trigonometric functions, so it should be remembered. We can also use similar property for other inverse trigonometric functions as, \[{{\csc }^{-1}}x+{{\sec }^{-1}}x=\dfrac{\pi }{2}\] and \[{{\tan }^{-1}}x+{{\cot }^{-1}}x=\dfrac{\pi }{2}\].

These properties are useful to solve other questions based on evaluation of expression, and proofs.

Recently Updated Pages

Basicity of sulphurous acid and sulphuric acid are

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

What is the stopping potential when the metal with class 12 physics JEE_Main

The momentum of a photon is 2 times 10 16gm cmsec Its class 12 physics JEE_Main

Using the following information to help you answer class 12 chemistry CBSE

Why should electric field lines never cross each other class 12 physics CBSE

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Describe the poetic devices used in the poem Aunt Jennifers class 12 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

And such too is the grandeur of the dooms We have imagined class 12 english CBSE

Where is the endless fountain of immortal drink pouring class 10 english CBSE

What I want should not be confused with total inactivity class 12 english CBSE

Why was the Moral Rearmament Army welcomed at the class 12 english CBSE

Difference Between Plant Cell and Animal Cell