Subjective Type

If $$\displaystyle \sin { \left( { \sin }^{ -1 }\frac { 1 }{ 5 } +{ \cos }^{ -1 }x \right) } =1$$, then find the value of $$x$$.

Solution

$$\displaystyle \sin { \left( { \sin }^{ -1 }\frac { 1 }{ 5 } +{ \cos }^{ -1 }x \right) } =1$$

$$\Rightarrow \displaystyle { \left( { \sin }^{ -1 }\frac { 1 }{ 5 } +{ \cos }^{ -1 }x \right) } =\sin^{-1}(1)=\frac{\pi}{2}$$

$$\Rightarrow \displaystyle \sin^{ -1 }\frac { 1 }{ 5 }=\frac{\pi}{2}-\cos^{-1}x=\sin^{-1}x$$

$$\displaystyle \therefore x=\frac{1}{5}$$

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