 Hello and welcome to the session. Let us understand the following problem. Prove that tan inverse 63 by 16 is equal to sin inverse 5 by 13 plus cos inverse 3 by 5. We will be using the following identity. tan of alpha plus beta is equal to tan alpha plus tan beta by 1 minus tan alpha into tan beta. This is our key idea. Now let us write the solution. We are given 5 by 13 and 3 by 5 as the inverse function of sin and cos on the right hand side. So let us first consider tan inverse 5 by 13. Let sin inverse of 5 by 13 is equal to alpha. Let us name it as 1. Therefore sin alpha is equal to 5 by 13. Now using Pythagoras theorem we get cos alpha is equal to 12 by 13 by Pythagoras theorem. Therefore tan alpha is equal to sin alpha by cos alpha which is equal to 5 by 12. Now let us consider cos inverse 3 by 5. Let cos inverse 3 by 5 is equal to beta. Let us name it as 2. Therefore cos beta is equal to 3 by 5. Now again by using Pythagoras theorem we get sin beta is equal to 4 by 5 by Pythagoras theorem. Therefore tan beta is equal to sin beta by cos beta which is equal to 4 by 3. Now using the identity that we stated in the key idea that is tan of alpha plus beta is equal to tan alpha plus beta upon 1 minus tan alpha into tan beta. Now substituting the values we get 5 by 12 plus 4 by 3 whole divided by 1 minus 5 by 12 into 4 by 3 which is equal to 15 plus 48 by 12 into 3 whole divided by 12 into 3 minus 20 whole divided by 12 into 3 which is equal to 63 by 36 whole divided by 16 by 36 which is equal to 63 by 16 thus tan of alpha plus beta is equal to 63 by 16 or alpha plus beta is equal to tan inverse of 63 by 16 or psi inverse of 5 by 13 plus cos inverse of 3 by 5 is equal to tan inverse of 63 by 16 by 1 and 2. Hence we got our desired result hence proved. I hope you understood this problem. Bye and have a nice day.