 Hello and welcome to the session. In this session we discussed the following question which says prove that sin inverse of 4 upon 5 plus sin inverse of 5 upon 13 plus sin inverse of 16 upon 65 is equal to pi by 2. Before we move on to the solution let's recall the results. Cos square x plus x is equal to 1 and cos of x plus y is equal to cos x into cos y minus sin x into sin y. This is the key idea that we use for this question. Let's proceed with the solution now. We have to prove sin inverse of 4 upon 5 plus sin inverse of 5 upon 13 plus sin inverse of 16 upon 65 is equal to pi by 2. This is what we are supposed to prove. First of all we take that x be equal to sin inverse of 4 upon 5. This gives us sin x is equal to 4 upon 5. Then we take let y be equal to sin inverse of 5 upon 13. This gives us sin y is equal to 5 upon 13. Now that we have got the values for sin x and sin y we find the values of cos x and cos y. Now we know that cos square x plus sin x is equal to 1. This means is equal to square root of 1 minus sin square x. Now putting the value for sin x as 4 upon 5 we get cos x is equal to square root of 1 minus 4 upon 5 the whole square. Which gives us cos x is equal to square root of 1 minus 16 upon 25 that is we get cos x is equal to square root of 25 minus 16 upon 25. This gives us cos x is equal to square root of 9 by 25. And this gives us cos x equal to 3 upon 5. Now similarly we get cos y is equal to square root of 1 minus sin square y. Now putting the value of sin y as 5 upon 13 we get cos y is equal to square root of 1 minus 5 upon 13 the whole square. Further cos y is equal to square root of 1 minus 25 upon 169 that is we have cos y is equal to square root of 169 minus 25 upon 169. Which gives us cos y is equal to square root of 144 upon 169 which means we have cos y is equal to 12 upon 13. This is the value for cos y. Now let's find out cos of x plus y which is equal to cos x into cos y minus sin x into sin y. Let's substitute the respective values. So we get cos of x plus y is equal to now the value for cos x is 3 upon 5. So this is equal to 3 upon 5 into cos y which is 12 upon 13 minus sin x. And the value for sin x is 4 upon 5 and value for sin y is 5 upon 13. So we have 4 upon 5 multiplied by 5 upon 13. Now this 5 and 5 gets cancelled and further we get cos of x plus y is equal to 36 upon 65 minus 4 upon 13 that is we get cos of x plus y is equal to 36 minus 20 upon 65. Now cos of x plus y is equal to 16 upon 65 or you can say we have x plus y is equal to cos inverse of 16 upon 65. Now earlier we had assumed x as sin inverse of 4 upon 5 and y as sin inverse of 5 upon 13. So we put the values of x plus y in this. So we get sin inverse of 4 upon 5 plus sin inverse of 5 upon 13 is equal to cos inverse of 16 upon 65. Now further we write this cos inverse in terms of sin inverse so we get sin inverse of 4 upon 5 plus sin inverse of 5 upon 13 is equal to pi by 2 minus sin inverse of 16 upon 65. Or you can say we get sin inverse of 4 upon 5 plus sin inverse of 5 upon 13 plus sin inverse of 16 upon 65 is equal to pi by 2. That is we have shifted this term to the left hand side so we get this and this is what we were supposed to prove so hence proved this completes the session hope you have understood the solution of this question.