Question

The value of \[{\cos ^{ - 1}}\left( {\cos 12} \right) - {\sin ^{ - 1}}\left( {\sin 14} \right)\] is:
A. \[ - 2\]
B. \[8\pi - 26\]
C. \[4\pi + 2\]
D. None of the above

Answer 1

Hint: In this question, we will proceed by using the formulae \[\cos \left( {4\pi - x} \right) = \cos x\] and \[\sin \left( {x - 4\pi } \right) = \sin x\]. Then we will further simplify the given expression further by using the formula \[{\cos ^{ - 1}}\left( {\cos x} \right) = x\] and \[{\sin ^{ - 1}}\left( {\sin x} \right) = x\].

Complete step by step answer:
Given expression is \[{\cos ^{ - 1}}\left( {\cos 12} \right) - {\sin ^{ - 1}}\left( {\sin 14} \right)...................\left( 1 \right)\]
We know that \[\cos \left( {4\pi - x} \right) = \cos x\] and \[\sin \left( {x - 4\pi } \right) = \sin x\]
By using these formulae in equation (1), we have
\[ \Rightarrow {\cos ^{ - 1}}\left( {\cos \left( {4\pi - 12} \right)} \right) - {\sin ^{ - 1}}\left( {\sin \left( {14 - 4\pi } \right)} \right)\]
Also, we know that \[{\cos ^{ - 1}}\left( {\cos x} \right) = x\] and \[{\sin ^{ - 1}}\left( {\sin x} \right) = x\]
By using these formulae in above expression, we get
\[
   \Rightarrow \left( {4\pi - 12} \right) - \left( {14 - 4\pi } \right) \\
   \Rightarrow 4\pi - 12 - 14 + 4\pi \\
  \therefore 8\pi - 26 \\
\]
Therefore, the value of the expression \[{\cos ^{ - 1}}\left( {\cos 12} \right) - {\sin ^{ - 1}}\left( {\sin 14} \right)\] is \[8\pi - 26\].

So, the correct answer is “Option B”.

Note: In order to solve this type of question one should think about inverse trigonometric functions. In mathematics, inverse trigonometric functions are also called arcus functions or anti-trigonometric functions are the inverse functions of the trigonometric functions. Specifically, they are inverses 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.