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If $y={{\cot }^{-1}}\left\{ \dfrac{\cos x-\sin x}{\cos x+\sin x} \right\}$ , find $\dfrac{dy}{dx}$ ?

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Last updated date: 13th Jun 2024
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Answer
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Hint: In this problem, we have to find the derivative of the given trigonometric function. So, we will apply the trigonometric formula and the derivation rules to get the solution. We start solving this problem, by dividing sinx on the RHS of the given function, then we will apply the trigonometric formula $\cot x=\dfrac{\cos x}{\sin x}$ . On further solving, we will again apply the trigonometric formula $\cot \left( x-\dfrac{\pi }{4} \right)=~\left\{ \dfrac{\cot x-\cot \dfrac{\pi }{4}}{\cot \dfrac{\pi }{4}\cot x+1} \right\}$ . As we know, the trigonometric formula ${{\cot }^{-1}}\left( \cot x \right)=x$ , thus we get the simplified function. After that, we will find its derivative, to get the required solution to the problem.

Complete step by step answer:
According to the question, we have to find the derivative of a function.
Thus, we will apply the trigonometric formula and the derivation rules to get the solution.
The function given to us is $y={{\cot }^{-1}}\left\{ \dfrac{\cos x-\sin x}{\cos x+\sin x} \right\}$ ----- (1)
Now, we will divide sinx on the RHS in the equation (1), we get
$y={{\cot }^{-1}}\left\{ \dfrac{\dfrac{\cos x}{\sin x}-\dfrac{\sin x}{\sin x}}{\dfrac{\cos x}{\sin x}+\dfrac{\sin x}{\sin x}} \right\}$
Now, we will apply the trigonometric formula $\cot x=\dfrac{\cos x}{\sin x}$ in the above equation, we get
$y={{\cot }^{-1}}\left\{ \dfrac{\cot x-1}{\cot x+1} \right\}$
Thus, we will again apply the trigonometric formula $\cot \left( x-\dfrac{\pi }{4} \right)=~\left\{ \dfrac{\cot x-\cot \dfrac{\pi }{4}}{\cot \dfrac{\pi }{4}\cot x+1} \right\}$ in the above equation, we get
$y={{\cot }^{-1}}\left\{ \cot \left( x-\dfrac{\pi }{4} \right) \right\}$
As we know, the trigonometric formula ${{\cot }^{-1}}\left( \cot x \right)=x$, thus we will substitute this value in the above equation, we get
$y=x-\dfrac{\pi }{4}$
Therefore, we get the simplified equation of the function. Now, we will find the derivative of the simplified function with respect to x, we get
$\dfrac{dy}{dx}=1$

Therefore, If $y={{\cot }^{-1}}\left\{ \dfrac{\cos x-\sin x}{\cos x+\sin x} \right\}$ , then its derivative with respect to x is equal to 1.

Note: While solving this problem, do mention all the formulas properly to avoid mathematical errors. One of the alternative methods to solve this problem is directly find the derivative of the given function with respect to x instead of finding its simplified value, to get the required solution for the problem.