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Question: If \[\sin \left( {{\sin }^{-1}}\left( \dfrac{1}{5} \right)+{{\cos }^{-1}}(x) \right)=1\]then x is eq...

If sin⁡(sin⁡−1(15)+cos⁡−1(x))=1\sin \left( {{\sin }^{-1}}\left( \dfrac{1}{5} \right)+{{\cos }^{-1}}(x) \right)=1then x is equal to
A. 11
B. 00
C.45\dfrac{4}{5}
D.15\dfrac{1}{5}

Explanation

Solution

Hint : In this particular problem, to find the value of sin⁡(sin⁡−1(15)+cos⁡−1(x))=1\sin \left( {{\sin }^{-1}}\left( \dfrac{1}{5} \right)+{{\cos }^{-1}}(x) \right)=1first of all we need take sin on RHS then becomessin⁡−1(1)=π2{{\sin }^{-1}}\left( 1 \right)=\dfrac{\pi }{2}. After that we have to simplify it and apply the formula that isπ2−sin⁡−1(15)=cos⁡−1(15)\dfrac{\pi }{2}-{{\sin }^{-1}}\left( \dfrac{1}{5} \right)={{\cos }^{-1}}\left( \dfrac{1}{5} \right). Then solve further and simplify it and get the values.

Complete step-by-step answer :
According to the question it is given that sin⁡(sin⁡−1(15)+cos⁡−1(x))=1\sin \left( {{\sin }^{-1}}\left( \dfrac{1}{5} \right)+{{\cos }^{-1}}(x) \right)=1
Before applying the formula first we need to simplify by taking on right hand side then it becomes sin⁡−1(1){{\sin }^{-1}}\left( 1 \right)therefore, sin⁡−1(1)=π2{{\sin }^{-1}}\left( 1 \right)=\dfrac{\pi }{2}substitute this value in above equation then we get:
sin⁡−1(15)+cos⁡−1(x)=sin⁡−1(1){{\sin }^{-1}}\left( \dfrac{1}{5} \right)+{{\cos }^{-1}}(x)={{\sin }^{-1}}\left( 1 \right)
Further solving and simplifying we get:
sin⁡−1(15)+cos⁡−1(x)=π2{{\sin }^{-1}}\left( \dfrac{1}{5} \right)+{{\cos }^{-1}}(x)=\dfrac{\pi }{2}
Here, if you notice this above equation you can see that there is only one variable to find and to simplify it further then we get:
cos⁡−1(x)=π2−sin⁡−1(15){{\cos }^{-1}}(x)=\dfrac{\pi }{2}-{{\sin }^{-1}}\left( \dfrac{1}{5} \right)
Now, we can apply the formula of π2−sin⁡−1(15)=cos⁡−1(15)\dfrac{\pi }{2}-{{\sin }^{-1}}\left( \dfrac{1}{5} \right)={{\cos }^{-1}}\left( \dfrac{1}{5} \right) and substitute this formula in this above equation we get:
cos⁡−1(x)=cos⁡−1(15){{\cos }^{-1}}(x)={{\cos }^{-1}}\left( \dfrac{1}{5} \right)
By comparing the cos⁡−1(x){{\cos }^{-1}}(x)and cos⁡−1(15){{\cos }^{-1}}\left( \dfrac{1}{5} \right)we get the get the value of x that is
x=15x=\dfrac{1}{5}
So, the correct option is “option D”.
So, the correct answer is “Option D”.

Note : In this problem always remember the formula which is used here in this problem. First of all this type of problem is reduced by taking sin on the right hand side. Don’t make any mistake while substituting or applying the formula. See the question, then reduce the problem if possible in the first step of this particular problem and further solve step by step and compare the values with trigonometry function and find the values of x, so, the above solution is preferred for such types of problems.