Question

# If $$2\tan ^{ -1 }{ \left( \cos { x } \right) } =\tan ^{ -1 }{ (2\text {cosec} { x } ) }$$, then $$\sin { x } +\cos { x } =$$

A
22
B
2
C
12
D
12

Solution

## The correct option is C $$\sqrt { 2 }$$$${ \tan }^{ -1 }(A)+{ \tan }^{ -1 }(B)={ \tan }^{ -1 }\left (\dfrac { A+B }{ 1-AB } \right)$$$$2{ \tan }^{ -1 }(\cos x)={ \tan }^{ -1 }(2\text {cosec} x)$$$$\Rightarrow$$ $${ \tan }^{ -1 }\left (\dfrac { 2\cos x }{ 1-\cos^{ 2 }x } \right)={ \tan }^{ -1 }\left (\dfrac { 2 }{ \sin x } \right)$$ $$\quad$$$$\quad$$$$\quad$$$$\quad$$$$\because$$$$\quad$$($${ \tan }^{ -1 }(A)+{ \tan }^{ -1 }(B)={ \tan }^{ -1 }\left (\dfrac { A+B }{ 1-AB } \right)$$) $$\Rightarrow$$ $$\dfrac { 2\cos x }{ \sin^{ 2 }x } =\dfrac { 2 }{ \sin x }$$ $$\Rightarrow$$$$\cos x=\sin x$$ $$\Rightarrow$$$$x={ 45 }^{ \circ }$$ $$\therefore$$ $$\cos x+\sin x=2\sin x=2\sin({ 45 }^{ \circ })=2\times \dfrac { 1 }{ \sqrt { 2 } } =\sqrt { 2 }$$ Mathematics

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