Question

Solve the equation for x: $\cos \left( {{\tan }^{-1}}x \right)=\sin \left( {{\cot }^{-1}}\dfrac{3}{4} \right)$

Answer 1

Hint: We will use the formula $\sin \left( \dfrac{\pi }{2}-x \right)=\cos x$ and the formula $\dfrac{\pi }{2}-{{\tan }^{-1}}x={{\cot }^{-1}}x$. First we will convert the cos into sin using the first formula. And then we will convert ${{\tan }^{-1}}$ to ${{\cot }^{-1}}$. And then we can see that both sides of the equation are equal, and with the help of that we will find the value of x.

Complete step-by-step answer:
Let’s start solving the question from LHS,
$\cos \left( {{\tan }^{-1}}x \right)$
Using the formula $\sin \left( \dfrac{\pi }{2}-x \right)=\cos x$ we get,
$\sin \left( \dfrac{\pi }{2}-{{\tan }^{-1}}x \right)$
Now using the formula $\dfrac{\pi }{2}-{{\tan }^{-1}}x={{\cot }^{-1}}x$ we get,
$\sin \left( {{\cot }^{-1}}x \right)$
Now we can see that the LHS = $\sin \left( {{\cot }^{-1}}x \right)$ and RHS = $\sin \left( {{\cot }^{-1}}\dfrac{3}{4} \right)$
Hence, from comparing we can say that $x=\dfrac{3}{4}$.
Hence, the question has been solved.

Note: We have only used the two trigonometric formula $\sin \left( \dfrac{\pi }{2}-x \right)=\cos x$ and the formula $\dfrac{\pi }{2}-{{\tan }^{-1}}x={{\cot }^{-1}}x$. These must be kept in mind. One can also solving this question by converting
${{\cot }^{-1}}\dfrac{3}{4}$ in the required angle and then finding it’s sin value. After that we can convert ${{\tan }^{-1}}$ to ${{\sin }^{-1}}$ in the LHS and then we can find the value of x. For that one must use the triangle to convert ${{\tan }^{-1}}$ to ${{\sin }^{-1}}$ and ${{\cot }^{-1}}\dfrac{3}{4}$ in the required angle. The value of ${{\cot }^{-1}}\dfrac{3}{4}$ is 53 degrees and it must be known to the student if he wishes to solve this question by using the second method.