 Hi, and welcome to our session. Let us discuss the following question. The question says, if x is equal to 2 cos t minus cos 2t, at y is equal to 2 sin t minus sin 2t, find the values of d2y by dx2 at t equals to pi by 2. Let's now begin with the solution. We are given that and y is equal to 2 sin t. We have to find the value of d2y by dx2 at t equals to pi by 2. Let's name this as equation number 1 and this as 2. Now differentiating 1 with respect to t, we get dx by dt equals to minus 2 sin t minus minus 2 sin 2t. This is equal to minus 2 sin t plus 2 sin 2t. Now we will again differentiate this equation that is dx by dt equals to minus 2 sin t plus 2 sin 2t. So we have dx by dt2 equals to minus 2 cos t plus 4 cos 2t. To differentiate 2 with respect to t, on differentiating 2 with respect to t, we get dy by dt equals to 2 cos t minus 2 cos 2t. Now we will again differentiate this equation. So we have d2y by dt2 equals to minus 2 sin t plus 4 sin 2t. Let us name this as equation number 1 and this as 2. Now from 1 and 2 we have d2y by dt2 divided by d2x by dt2 equals to minus 2 sin t plus 4 sin 2t. By minus 2 cos t plus 4 cos 2t. This implies d2y by d2x is equal to 4 sin 2t minus 2 sin t divided by 4 cos 2t minus 2 cos t and this is equal to 2 sin 2t minus sin t divided by 2 cos 2t minus cos t. Now we will find value of d2y by dx2 as x equals to pi by 2. So this is equal to 2 sin 2 into pi by 2 minus sin pi divided by 2 cos 2 into pi by 2 minus cos pi by 2. This is equal to 0 minus 0 divided by minus 1 minus 0 and this is equal to 0. So value of d2y by dx2 and x equals to pi by 2 is 0. This is our required answer. So this completes the session. Bye and take care.