 Hello and welcome to the session. In this session we are going to discuss the following question which says that prove that cos inverse of 5 by 13 minus sin inverse of 4 by 5 is equal to 5 minus sin inverse of 16 by 65. We know that sin inverse of x minus sin inverse of y is equal to pi minus sin inverse of x into square root of 1 minus y square minus y into square root of 1 minus x square. If both x and y are greater than equal to 0 and x square plus y square is greater than 1, with this key idea we shall proceed with the solution. We need to prove that cos inverse of 5 by 13 minus sin inverse of 4 by 5 is equal to 5 minus sin inverse of 16 by 65. Taking the left hand side of the expression we have cos inverse of 5 by 13 minus sin inverse of 4 by 5 first let us take cos inverse of 5 by 13 d theta which implies that 5 by 13 is equal to cos theta. All we can write cos theta is equal to 5 by 13 in a triangle ABC if theta is the angle between the lines AC and CB. We know that cos of theta is given by base upon hypotenuse that is BC upon AC which is equal to 5 by 13 so we have the value of the base BC as 5 and the value of the hypotenuse AC as 13 then we can find the value of the prependicular AB by using Pythagoras theorem. By Pythagoras theorem we have prependicular AB is equal to square root of hypotenuse square that is AC square minus base square that is BC square which is equal to square root of 13 square minus 5 square which is further equal to square root of 169 minus 25 which is given by square root of 144 equal to 12. Therefore we have value of the prependicular AB as 12. Now we know that sin of angle theta is given by prependicular upon hypotenuse that is AB upon AC which is equal to 12 by 13 which implies that theta is equal to sin inverse of 12 by 13 Now we have cos inverse of 5 by 13 is equal to theta and theta is equal to sin inverse of 12 by 13 which implies that cos inverse of 5 by 13 is equal to sin inverse of 12 by 13 Now replace cos inverse of 5 by 13 with sin inverse of 12 by 13 in the left hand side of the given expression therefore the left hand side becomes sin inverse of 12 by 13 minus sin inverse of 4 by 5 Answer and the key idea we know that sin inverse of x minus sin inverse of y is equal to pi minus sin inverse of x into square root of 1 minus y square minus y into square root of 1 minus x square provided both x and y are greater than 0 and x square plus y square is greater than 1 Here we assume x is 12 by 13 and y is 4 by 5 and we know that 12 by 13 is greater than 0 also 4 by 5 is greater than 0 and square root of 12 by 13 plus square root of 4 by 5 is greater than 1 Therefore sin inverse of 12 by 13 minus sin inverse of 4 by 5 can be written as pi minus sin inverse of 12 by 13 into square root of 1 minus 4 by 5 square minus 4 by 5 into square root of 1 minus y square minus square root of 4 by 5 sin inverse of 1 minus 12 by 13 square that is pi minus sin inverse of 12 by 13 into square root of 1 minus 16 by 25 minus 4 by 5 into square root of 1 minus 144 upon 169 This can be written as pi minus sin inverse of 12 by 13 into square root of 25 minus 16 by 25 minus 4 by 5 square root of 169 minus 144 by 169 Thus we get pi minus sin inverse of 12 by 13 into square root of 9 by 25 minus 4 by 5 into square root of 25 upon 169 Which is equal to pi minus sin inverse of 12 by 13 into 3 by 5 minus 4 by 5 into 5 upon 13 which further gives pi minus sin inverse of 36 upon 65 minus 20 upon 65 Which is equal to pi minus sin inverse of 16 upon 65 so we get sin inverse of 12 by 13 minus sin inverse of 4 by 5 as pi minus sin inverse of 16 by 65 which is equal to the right hand side Or we can also write it as sin inverse of 5 by 13 minus sin inverse of 4 by 5 is equal to pi minus sin inverse of 16 by 65 Thus we have proved our result that this sin inverse of 5 by 13 minus sin inverse of 4 by 5 is equal to pi minus sin inverse of 16 by 65 This completes our session hope you enjoyed this session