Question

If \[\sin \left( {{\sin }^{-1}}\left( \dfrac{1}{5} \right)+{{\cos }^{-1}}(x) \right)=1\]then x is equal to
A. \[1\]
B. \[0\]
C.\[\dfrac{4}{5}\]
D.\[\dfrac{1}{5}\]

Answer 1

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

Complete step-by-step answer:
According to the question it is given that \[\sin \left( {{\sin }^{-1}}\left( \dfrac{1}{5} \right)+{{\cos }^{-1}}(x) \right)=1\]
Before applying the formula first we need to simplify by taking on right hand side then it becomes \[{{\sin }^{-1}}\left( 1 \right)\]therefore, \[{{\sin }^{-1}}\left( 1 \right)=\dfrac{\pi }{2}\]substitute this value in above equation then we get:
\[{{\sin }^{-1}}\left( \dfrac{1}{5} \right)+{{\cos }^{-1}}(x)={{\sin }^{-1}}\left( 1 \right)\]
Further solving and simplifying we get:
\[{{\sin }^{-1}}\left( \dfrac{1}{5} \right)+{{\cos }^{-1}}(x)=\dfrac{\pi }{2}\]
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:
\[{{\cos }^{-1}}(x)=\dfrac{\pi }{2}-{{\sin }^{-1}}\left( \dfrac{1}{5} \right)\]
Now, we can apply the formula of \[\dfrac{\pi }{2}-{{\sin }^{-1}}\left( \dfrac{1}{5} \right)={{\cos }^{-1}}\left( \dfrac{1}{5} \right)\] and substitute this formula in this above equation we get:
\[{{\cos }^{-1}}(x)={{\cos }^{-1}}\left( \dfrac{1}{5} \right)\]
By comparing the \[{{\cos }^{-1}}(x)\]and \[{{\cos }^{-1}}\left( \dfrac{1}{5} \right)\]we get the get the value of x that is
\[x=\dfrac{1}{5}\]
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.