Question

Assertion :Let $f\left(x\right)=co{t}^{-1}\left(\frac{3x-x^3}{1-3x^2}\right),g\left(x\right)=co{s}^{-1}\left(\frac{1-x^2}{1+x^2}\right)$ then $f\left(x\right)=g\left(x\right)$ if $x=\frac{\sqrt{25-10\sqrt{5}}}{5}$. Reason: $sin{18}^o=\frac{\sqrt{5}-1}{4}$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

$f\left(x\right)=g\left(x\right){cot}^{-1}\left(\frac{3x-x^3}{1-3x^2}\right)={cos}^{-1}\left(\frac{1-x^2}{1+x^2}\right)={cot}^{-1}\left(\frac{1-x^2}{2x}\right)\Rightarrow \frac{3x-x^3}{1-3x^2}=\frac{1-x^2}{2x}\Rightarrow 5x^4-10x^2+1=0\Rightarrow x^2=\frac{10\pm 4\sqrt{5}}{10}=\frac{5\pm 2\sqrt{5}}{5}\Rightarrow x=\sqrt{\frac{5\pm 2\sqrt{5}}{5}}=\frac{\sqrt{25\pm 5\sqrt{5}}}{5}$