Find the principal value of \[{\cos ^{ - 1}}\left\{ {\left( {\dfrac{1}{{\sqrt 2 }}} \right)\left[ {\cos \left( {\dfrac{{9\pi }}{{10}}} \right) - \sin \left( {\dfrac{{9\pi }}{{10}}} \right)} \right]} \right\}\].
A.\[\dfrac{{3\pi }}{{10}}\]
B. \[\dfrac{{17\pi }}{{10}}\]
C. \[\dfrac{{7\pi }}{{10}}\]
D. None of these

Hints Express \[\dfrac{{9\pi }}{{10}}\] as\[\left( {\pi - \dfrac{\pi }{{10}}} \right)\] in the given expression and calculate. Substitute \[\dfrac{1}{{\sqrt 2 }}\] as \[\cos \dfrac{\pi }{4}\] or \[\sin \dfrac{\pi }{4}\] as required in the obtained expression. Then use the sum formula of cosine and calculate to obtain the required result.

Formula used
\[\cos (\pi - x) = - \cos x\]
\[\sin (\pi - x) = \sin x\]
\[{\cos ^{ - 1}}\left[ {\cos ( - x)} \right] = \pi - x\]
\[\cos (A - B) = \cos A\cos B + \sin A\sin B\]

Complete step by step solution
The given expression is,
\[{\cos ^{ - 1}}\left\{ {\left( {\dfrac{1}{{\sqrt 2 }}} \right)\left[ {\cos \left( {\dfrac{{9\pi }}{{10}}} \right) - \sin \left( {\dfrac{{9\pi }}{{10}}} \right)} \right]} \right\}\]
Express \[\dfrac{{9\pi }}{{10}}\] as\[\left( {\pi - \dfrac{\pi }{{10}}} \right)\] in the given expression and calculate.
\[ = {\cos ^{ - 1}}\left\{ {\left( {\dfrac{1}{{\sqrt 2 }}} \right)\left[ {\cos \left( {\pi - \dfrac{\pi }{{10}}} \right) - \sin \left( {\pi - \dfrac{\pi }{{10}}} \right)} \right]} \right\}\]
Use the formula \[\cos (\pi - x) = - \cos x\] and \[\sin (\pi - x) = \sin x\],
\[ = {\cos ^{ - 1}}\left\{ {\left( {\dfrac{1}{{\sqrt 2 }}} \right)\left[ { - \cos \left( {\dfrac{\pi }{{10}}} \right) - \sin \left( {\dfrac{\pi }{{10}}} \right)} \right]} \right\}\]
\[ = {\cos ^{ - 1}}\left\{ { - \dfrac{1}{{\sqrt 2 }}.\cos \left( {\dfrac{\pi }{{10}}} \right) - \dfrac{1}{{\sqrt 2 }}.\sin \left( {\dfrac{\pi }{{10}}} \right)} \right\}\]
Substitute \[\dfrac{1}{{\sqrt 2 }}\] as \[\cos \dfrac{\pi }{4}\] or \[\sin \dfrac{\pi }{4}\] as required in the obtained expression.
\[ = {\cos ^{ - 1}}\left\{ { - \left\{ {\cos \dfrac{\pi }{4}.\cos \left( {\dfrac{\pi }{{10}}} \right) + \sin \dfrac{\pi }{4}.\sin \left( {\dfrac{\pi }{{10}}} \right)} \right\}} \right\}\]
\[ = {\cos ^{ - 1}}\left\{ { - \left\{ {\cos \dfrac{\pi }{4}.\cos \left( {\dfrac{\pi }{{10}}} \right) + \sin \dfrac{\pi }{4}.\sin \left( {\dfrac{\pi }{{10}}} \right)} \right\}} \right\}\]
Use the formula \[\cos (A - B) = \cos A\cos B + \sin A\sin B\] for further calculation.
\[ = {\cos ^{ - 1}}\left\{ { - \cos \left( {\dfrac{\pi }{4} - \dfrac{\pi }{{10}}} \right)} \right\}\]
\[ = \pi - {\cos ^{ - 1}}\left\{ {\cos \left( {\dfrac{{3\pi }}{{20}}} \right)} \right\}\]
\[ = \pi - \dfrac{{3\pi }}{{20}}\]
\[ = \dfrac{{17\pi }}{{20}}\]
The Correct option is B

Note To solve this type of problem students must aware of all the formula that are used in this problem, there are a lot of formulas used in this problem. If any one of the formulas is not applied properly then the we will not be able to solve the problem.