 Hello and welcome to the session. In this session we discussed the following question. Prove the following, tan inverse 1 upon 3 plus tan inverse 1 upon 5 plus tan inverse 1 upon 7 plus tan inverse 1 upon 8 is equal to pi by 4. Let's recall the formula, tan inverse x plus tan inverse y is equal to tan inverse x plus y upon 1 minus xy, where xy is less than 1. This is the key idea to be used for this question. Now let's move on to the solution. We have to prove that tan inverse 1 upon 3 plus tan inverse 1 upon 5 plus tan inverse 1 upon 7 plus tan inverse 1 upon 8 is equal to pi upon 4. Now first let's consider the LHS that is tan inverse 1 upon 3 plus tan inverse 1 upon 5 plus tan inverse 1 upon 7 plus tan inverse 1 upon 8. We put these two terms in one bracket and these two terms in the other bracket. Now we can apply the formula that we had written in the key idea of tan inverse x plus tan inverse y in the first bracket and in the second bracket also. So on applying this formula this would be equal to tan inverse x plus y that is 1 upon 3 plus 1 upon 5 upon 1 minus 1 upon 3 into 1 upon 5 plus tan inverse 1 upon 7 plus 1 upon 8 upon 1 minus 1 upon 7 into 1 upon 8. This further is equal to tan inverse 8 upon 15 upon 1 minus 1 upon 15 plus tan inverse 15 upon 56 upon 1 minus 1 upon 56. This further is equal to tan inverse 8 upon 15 upon 15 minus 1 upon 15 plus tan inverse 15 upon 56 upon 56 minus 1 upon 56. So this is equal to tan inverse 8 upon 15 upon 14 upon 15 plus tan inverse 15 upon 56 upon 55 upon 56. That we get tan inverse 8 upon 14 plus tan inverse 15 upon 55. This further is equal to tan inverse 4 upon 7 plus tan inverse 3 upon 11. Now here we can apply the formula for tan inverse x plus tan inverse y. So this would be equal to tan inverse x plus y that is 4 upon 7 plus 3 upon 11 upon 1 minus x into y that is 4 upon 7 into 3 upon 11. This further is equal to tan inverse 44 plus 21 upon 77 whole upon 1 minus 12 upon 77. This becomes equal to tan inverse 65 upon 77 whole upon 77 minus 12 upon 77 that is this is equal to tan inverse 65 upon 77 upon 65 upon 77. So we have LHS is equal to tan inverse 1 and this is equal to pi upon 4 which is equal to the RHS. Thus we get LHS is equal to the RHS that is we have tan inverse 1 upon 3 plus tan inverse 1 upon 5 plus tan inverse 1 upon 7 plus tan inverse 1 upon 8 is equal to pi upon 4. Hence proved. So this completes the session. Hope you have understood the solution for this question.