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Question

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


A
22
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B
2
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C
12
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D
12
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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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