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Question: Prove that the derivative is \(\dfrac{{dy}}{{dx}} = \dfrac{1}{2}\) , when \(y = {\tan ^{ - 1}}\dfrac...

Prove that the derivative is dydx=12\dfrac{{dy}}{{dx}} = \dfrac{1}{2} , when y=tan⁡−1sin⁡x1+cos⁡xy = {\tan ^{ - 1}}\dfrac{{\sin x}}{{1 + \cos x}}

Explanation

Solution

We have to prove that dydx=12\dfrac{{dy}}{{dx}} = \dfrac{1}{2} and for that we have to find the value of dydx\dfrac{{dy}}{{dx}} and for that we will find the derivative of the function given in the question. We will differentiate the function with respect to ‘x’ and by using the appropriate formulas of trigonometric functions.

Complete step-by-step solution:
Given: y=tan⁡−1sin⁡x1+cos⁡xy = {\tan ^{ - 1}}\dfrac{{\sin x}}{{1 + \cos x}}
By using the formula sin⁡x=2sin⁡x2cos⁡x2\sin x = 2\sin \dfrac{x}{2}\cos \dfrac{x}{2} and cos⁡x=cos⁡2x2−sin⁡2x2\cos x = {\cos ^2}\dfrac{x}{2} - {\sin ^2}\dfrac{x}{2} we will rewrite the function given above.
y=tan⁡−12sin⁡x2cos⁡x21+cos⁡2x2−sin⁡2x2y = {\tan ^{ - 1}}\dfrac{{2\sin \dfrac{x}{2}\cos \dfrac{x}{2}}}{{1 + {{\cos }^2}\dfrac{x}{2} - {{\sin }^2}\dfrac{x}{2}}}
By using the trigonometric formula we can write cos⁡2x2−sin⁡2x2{\cos ^2}\dfrac{x}{2} - {\sin ^2}\dfrac{x}{2} as 2cos⁡2x2−12{\cos ^2}\dfrac{x}{2} - 1 .
So, replacing it in the denominator.
y=tan⁡−12sin⁡x2cos⁡x21+2cos⁡2x2−1y = {\tan ^{ - 1}}\dfrac{{2\sin \dfrac{x}{2}\cos \dfrac{x}{2}}}{{1 + 2{{\cos }^2}\dfrac{x}{2} - 1}}
y=tan⁡−12sin⁡x2cos⁡x22cos⁡2x2y = {\tan ^{ - 1}}\dfrac{{2\sin \dfrac{x}{2}\cos \dfrac{x}{2}}}{{2{{\cos }^2}\dfrac{x}{2}}}
Now, we will take 2cos⁡x22\cos \dfrac{x}{2} common from numerator and denominator and cut them from each other.
y=tan⁡−1sin⁡x2cos⁡x2y = {\tan ^{ - 1}}\dfrac{{\sin \dfrac{x}{2}}}{{\cos \dfrac{x}{2}}}
By using the trigonometric formula sin⁡Acos⁡A=tan⁡A\dfrac{{\sin A}}{{\cos A}} = \tan A we will rewrite the function.
y=tan⁡−1(tan⁡x2)y = {\tan ^{ - 1}}\left( {\tan \dfrac{x}{2}} \right)
Now, according to trigonometric propertiestan⁡−1(tan⁡A){\tan ^{ - 1}}\left( {\tan A} \right) is equal to AA . So, replacing it in the equation above.
y=x2y = \dfrac{x}{2}
We will now differentiate the equation above with respect to ‘x’ to find the value of dydx\dfrac{{dy}}{{dx}} .
dydx=12\dfrac{{dy}}{{dx}} = \dfrac{1}{2}
We know that the derivative of xx is 11. So, the derivative of x2\dfrac{x}{2} will be 12\dfrac{1}{2} .
Hence, proved.

Note: Differentiation is a method of finding the derivative of function. Differentiation is a process, in Maths, where we find the instantaneous rate of change in function based on one of its variables. The most common example is the rate change of displacement with respect to time, called velocity. The opposite of finding a derivative is anti-differentiation.