Question

Find the correct option.
The value of \[\sin [{\sin ^{ - 1}}(\dfrac{1}{3}) + {\sec ^{ - 1}}(3)] + \cos [{\tan ^{ - 1}}(\dfrac{1}{2}) + {\tan ^{ - 1}}(2)]\] is
A. 1
B. 2
C. 3
D. 4

Answer 1

Hint: The subparts of the given expression will be first defined and using relevant inverse trigonometric identities, those subparts will be simplified and then putting the sine and cosine values of the simplified angles, the final value of the expression will be calculated and the correct option will be chosen.

Formulae Used: The following inverse trigonometric identities will be used to simplify the given expression:
1. \[{\sec ^{ - 1}}(x) = {\cos ^{ - 1}}(\dfrac{1}{x})\]
2. \[{\tan ^{ - 1}}(x) = {\cot ^{ - 1}}(\dfrac{1}{x})\]
3. \[{\sin ^{ - 1}}x + {\cos ^{ - 1}}x = \dfrac{\pi }{2}\]
4. \[{\tan ^{ - 1}}x + {\cot ^{ - 1}}x = \dfrac{\pi }{2}\]

Complete step-by-step solution:
We have been given the expression \[\sin [{\sin ^{ - 1}}(\dfrac{1}{3}) + {\sec ^{ - 1}}(3)] + \cos [{\tan ^{ - 1}}(\dfrac{1}{2}) + {\tan ^{ - 1}}(2)]\].
We have to simplify the above expression and find its value to choose the correct option.
Let, \[y = \sin [{\sin ^{ - 1}}(\dfrac{1}{3}) + {\sec ^{ - 1}}(3)] + \cos [{\tan ^{ - 1}}(\dfrac{1}{2}) + {\tan ^{ - 1}}(2)]\]
We will define the sub-parts of the expression as follows:
Let, \[{y_1} = \sin [{\sin ^{ - 1}}(\dfrac{1}{3}) + {\sec ^{ - 1}}(3)]\]
And \[{y_2} = \cos [{\tan ^{ - 1}}(\dfrac{1}{2}) + {\tan ^{ - 1}}(2)]\]
So, \[y = {y_1} + {y_2}\]
First, we will simplify \[{y_1} = \sin [{\sin ^{ - 1}}(\dfrac{1}{3}) + {\sec ^{ - 1}}(3)]\] as follows:
We know that \[{\sec ^{ - 1}}(x) = {\cos ^{ - 1}}(\dfrac{1}{x})\]
Substituting \[x\] with \[3\], we have
\[{\sec ^{ - 1}}(3) = {\cos ^{ - 1}}(\dfrac{1}{3})\]
So, \[{y_1} = \sin [{\sin ^{ - 1}}(\dfrac{1}{3}) + {\sec ^{ - 1}}(3)]\]
\[ = \sin [{\sin ^{ - 1}}(\dfrac{1}{3}) + {\cos ^{ - 1}}(\dfrac{1}{3})]\] [Since \[{\sec ^{ - 1}}(3) = {\cos ^{ - 1}}(\dfrac{1}{3})\]]
Now, using the formula \[{\sin ^{ - 1}}x + {\cos ^{ - 1}}x = \dfrac{\pi }{2}\] and substituting \[x\] with \[\dfrac{1}{3}\], we have
\[{y_1} = \sin [{\sin ^{ - 1}}(\dfrac{1}{3}) + {\cos ^{ - 1}}(\dfrac{1}{3})]\]
\[ = \sin \dfrac{\pi }{2}\]
Similarly, we will simplify \[{y_2} = \cos [{\tan ^{ - 1}}(\dfrac{1}{2}) + {\tan ^{ - 1}}(2)]\] as follows:
We know \[{\tan ^{ - 1}}(x) = {\cot ^{ - 1}}(\dfrac{1}{x})\]
Substituting \[x\] with \[\dfrac{1}{2}\], we have
\[{\tan ^{ - 1}}(\dfrac{1}{2}) = {\cot ^{ - 1}}(2)\] [Since \[\dfrac{1}{{\dfrac{1}{2}}} = 1 \times \dfrac{2}{1} = 2\]]
So, \[{y_2} = \cos [{\tan ^{ - 1}}(\dfrac{1}{2}) + {\tan ^{ - 1}}(2)]\]
\[ = \cos [{\cot ^{ - 1}}(2) + {\tan ^{ - 1}}(2)]\] [Since \[{\tan ^{ - 1}}(\dfrac{1}{2}) = {\cot ^{ - 1}}(2)\] ]
\[ = \cos [{\tan ^{ - 1}}(2) + {\cot ^{ - 1}}(2)]\]
Now, using the formula \[{\tan ^{ - 1}}x + {\cot ^{ - 1}}x = \dfrac{\pi }{2}\] and substituting \[x\] with \[2\], we have
\[{y_2} = \cos [{\tan ^{ - 1}}(2) + {\cot ^{ - 1}}(2)]\]
\[ = \cos \dfrac{\pi }{2}\]
Now, assigning the sine and cosine values of the angle \[\dfrac{\pi }{2}\] , we will find the value of \[y\] as follows:
\[y = {y_1} + {y_2}\]
\[ = \sin \dfrac{\pi }{2} + \cos \dfrac{\pi }{2}\] [Since \[\sin \dfrac{\pi }{2} = 1\] and \[\cos \dfrac{\pi }{2} = 0\] ]
\[ = 1 + 0\]
\[ = 1\]
So, the value of \[\sin [{\sin ^{ - 1}}(\dfrac{1}{3}) + {\sec ^{ - 1}}(3)] + \cos [{\tan ^{ - 1}}(\dfrac{1}{2}) + {\tan ^{ - 1}}(2)]\] is equal to \[1\] .

Hence, option A. 1 is the correct answer.

Note: The inverse trigonometric functions are the inverse functions of the trigonometric functions. Especially, those are the inverse functions of the sine, cosine, tangent, cotangent, secant and cosecant functions and are used to obtain an angle from any of the angle’s trigonometric ratios. The inverse symbol, for example, \[{\sin ^{ - 1}}x\] should not be confused with \[{(\sin x)^{ - 1}}\] .