Answer

Verified

380.4k+ 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

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

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

How many crores make 10 million class 7 maths CBSE

The 3 + 3 times 3 3 + 3 What is the right answer and class 8 maths CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Change the following sentences into negative and interrogative class 10 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE