Given that the inverse trigonometric functions take principal values only, the number of real values of x which satisfy sin−153x+sin−154x=sin−1x is equal to
Solutions · 1
Official solution
\[{\sin ^{ - 1}}{{3x} \over 5} + {\sin ^{ - 1}}{{4x} \over 5} = {\sin ^{ - 1}}x\]
\[{\sin ^{ - 1}}\left( {{{3x} \over 5}\sqrt {1 - {{16{x^2}} \over {25}}} + {{4x} \over 5}\sqrt {1 - {{9{x^2}} \over {25}}} } \right) = {\sin ^{ - 1}}x\]
\[{{3x} \over 5}\sqrt {1 - {{16{x^2}} \over {25}}} + {{4x} \over 5}\sqrt {1 - {{9{x^2}} \over {25}}} = x\]
x=0 or \[3\sqrt {25 - 16{x^2}} + 4\sqrt {25 - 9{x^2}} = 25\]
\[4\sqrt {25 - 9{x^2}} = 25 - 3\sqrt {25 - 16{x^2}} \]
Squaring we get
\[16(25 - 9{x^2}) = 625 - 9(25 - 16{x^2}) - 150\sqrt {25 - 16{x^2}} \]
\[400 = 625 + 225 - 150\sqrt {25 - 16{x^2}} \]
\[\sqrt {25 - 16{x^2}} = 3 \Rightarrow 25 - 16{x^2} = 9\]
\[ \Rightarrow {x^2} = 1\]
Put x = 0, 1, -1 in the original equation
We see that all values satisfy the original equation.
Number of solution = 3