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Question: Find the value of \[\cos \left( {{\sec }^{-1}}x+{{\operatorname{cosec}}^{-1}}x \right)\]\[\left| x \...

Find the value of \cos \left( {{\sec }^{-1}}x+{{\operatorname{cosec}}^{-1}}x \right)$$$$\left| x \right|\ge 1.

Explanation

Solution

Hint: In this question, we first need to find the value of the term present in the bracket. Now, from the properties of inverse trigonometric functions we have that sec⁡−1x+cosec⁡−1x=π2{{\sec }^{-1}}x+{{\operatorname{cosec}}^{-1}}x=\dfrac{\pi }{2}. Then using the trigonometric ratios of some standard angles we can get the cosine value of the angle so obtained which gives the final result.

Complete step-by-step answer:
INVERSE FUNCTION:
If y=f(x)y=f\left( x \right) and x=g(y)x=g\left( y \right)are two functions such that f(g(y))=yf\left( g\left( y \right) \right)=yand g(f(y))=xg\left( f\left( y \right) \right)=x, then f and y are said to be inverse of each other
x=f−1(y)x={{f}^{-1}}\left( y \right)
INVERSE TRIGONOMETRIC FUNCTIONS:
As we know that trigonometric functions are not one-one and onto in their natural domain and range, so their inverse do not exist but if we restrict their domain and range, then their inverse may exist.
Here, from the properties of inverse trigonometric functions we have that
sec⁡−1x+cosec⁡−1x=π2{{\sec }^{-1}}x+{{\operatorname{cosec}}^{-1}}x=\dfrac{\pi }{2} if ∣x∣≥1\left| x \right|\ge 1
Now, from the given function in the question we have that
⇒cos⁡(sec⁡−1x+cosec⁡−1x)\Rightarrow \cos \left( {{\sec }^{-1}}x+{{\operatorname{cosec}}^{-1}}x \right)
Let us now substitute the respective value from the property of inverse trigonometric function
sec⁡−1x+cosec⁡−1x=π2{{\sec }^{-1}}x+{{\operatorname{cosec}}^{-1}}x=\dfrac{\pi }{2}
Now, on substituting this value in the above given function we get,
⇒cos⁡(π2)\Rightarrow \cos \left( \dfrac{\pi }{2} \right)
Here, as we already know that from the trigonometric ratios of some standard angles that
cos⁡π2=0\cos \dfrac{\pi }{2}=0
Now, on substituting this value in the above expression we get,
⇒0\Rightarrow 0
Hence, the value of cos⁡(sec⁡−1x+cosec⁡−1x)\cos \left( {{\sec }^{-1}}x+{{\operatorname{cosec}}^{-1}}x \right) in the given domain is 0.

Note: Instead of directly using the property of inverse trigonometric functions we can also solve it by converting them into inverse of sine and cosine functions and then using the respective property to simplify further. Both the methods give the same result.It is important to note that while solving the question we need to consider the respective property and also check for the domain which is given because the value of the functions changes accordingly.