 Hello and welcome to the session. Let's discuss the following question. It says differentiate tan inverse of under the root 1 plus x minus under the root 1 minus x upon under the root 1 plus x plus under the root 1 minus x with respect to x. So let's now move on to the solution and let y be equal to tan inverse under the root 1 plus x minus under the root 1 minus x upon under the root 1 plus x plus under the root 1 minus x. Now put x is equal to cos 2 theta. So we have y is equal to tan inverse under the root 1 plus cos 2 theta minus under the root 1 minus cos 2 theta upon under the root 1 plus cos 2 theta plus under the root 1 minus cos 2 theta. Now again y is equal to tan inverse 1 plus cos 2 theta is cos is 2 cos square theta. So we have 1 plus cos 2 theta is 2 cos square theta minus 1 minus cos 2 theta is 2 sin square theta upon under the root 2 cos square theta plus under the root 2 sin square theta. You need to remember this formulae and this is very important to write. So now y is equal to tan inverse root 2 cos theta minus root 2 sin theta upon root 2 cos theta plus root 2 sin theta. So y is equal to tan inverse cos theta minus sin theta upon cos theta plus sin theta. Now here dividing both numerator and denominator by cos theta we have y is equal to tan inverse 1 minus tan theta upon 1 plus tan theta. Now again y can be written as tan inverse 1 can be written as tan pi by 4 and here we have tan theta upon tan pi by 4 plus tan theta. So now again y is equal to tan inverse tan pi by 4 minus theta as we know that tan A minus B is equal to tan A minus tan B upon tan A plus tan B. So y is equal to pi by 4 minus theta. Now we assume that x is equal to cos 2 theta. So this implies 2 theta is equal to cos inverse x. So this implies theta is equal to 1 by 2 cos inverse x. So y is equal to pi by 4 minus 1 by 2 cos inverse x. So now we find dy by dx pi by 4 is a constant so its derivative is 0 and the derivative of cos inverse x is derivative of cos inverse x is minus 1 upon under the root 1 minus x square. So this is equal to minus minus plus and this becomes equal to 1 upon 2 into under the root 1 minus x square. Hence the answer is 1 upon 2 into under the root 1 minus x square. So this completes the question and the session. Bye for now. Take care and have a good day.