 Hello and welcome to the session. In this session we discussed the following question that says, who is the following? Cos inverse of 12 upon 13 plus sin inverse of 3 upon 5 is equal to sin inverse of 56 upon 65. Before moving on to the solution, let's discuss the formula for sin of a plus b. This is equal to sin a into cos b plus cos a into sin b. This is the key idea that we use in this question. Let's move on to the solution now. We need to prove that cos inverse of 12 upon 13 plus sin inverse of 3 upon 5 is equal to sin inverse of 56 upon 65. For this, first of all we suppose let cos inverse of 12 upon 13 be equal to y. This means that cos y is equal to 12 upon 13. When I just result 1, now we know that sin y is equal to square root of 1 minus cos square y. Since we know that sin square x plus cos square x is equal to 1. Now putting the value of cos y as 12 upon 13 in this, we get sin y is equal to square root of 1 minus 12 upon 13 the whole square. Further we have sin y is equal to square root of 169 minus 144 upon 169 that is we have sin y is equal to square root of 25 upon 169 which is equal to 5 upon 13. That is we now have the value of sin y as 5 upon 13. Let this be result 2. Well next we suppose the sin inverse of 3 upon 5 that is let sin inverse of 3 upon 5 be equal to z. So this means sin z is equal to 3 upon 5. Let this be result 3. Now again cos z would be equal to square root of 1 minus sin square z. Putting the value of sin z as 3 upon 5 in this we get cos z is equal to square root of 1 minus 3 upon 5 the whole square. This gives us cos z is equal to square root of 25 minus 9 upon 25 which gives us cos z is equal to square root of 16 upon 25 which is equal to 4 upon 5. Thus we have the value of cos z as 4 upon 5. Let this be result 4. Now y plus z would be equal to sin y into cos z plus cos y into sin z. Now we have the values for all sin y cos z cos y sin z. So putting the respective values we get sin of y plus z is equal to sin y which is equal to 5 upon 13. So putting sin y as sin upon 13 in this formula of sin of y plus z this into cos z which is equal to 4 upon 5. So here we put 4 upon 5 plus cos y which is equal to 12 upon 13. So here we put 12 upon 13 into sin z which is equal to 3 upon 5 this gives us y plus z is equal to 20 upon 65 plus 36 upon which further gives us y plus z is equal to 56 upon 6. This means that y plus z is equal to sin inverse of 56 upon 65 and we had assumed z to be equal to sin inverse of 3 upon 5 and y to be equal to cos inverse of 12 upon 13. So putting these values of y and z here we get cos inverse of 12 upon 13 plus n inverse of 3 upon 5 is equal to sin inverse of 56 upon 65. And this is what we were supposed to prove. So we have cos inverse of 12 upon 13 plus sin inverse of 3 upon 5 is equal to sin inverse of 56 upon 65. Hence the PhD session provides an understanding solution of this question.