 Hello and welcome to the session. In this session, we are going to discuss the following question which says that Evaluate the following First, coset inverse of coset of 5 pi by 6 Second, cos inverse of coset of 5 pi by 4 Third, sin of second inverse of x Fourth, cos of tan inverse of 2 by 3 Let's start the solution We have coset inverse of coset of 5 pi by 6 We know that coset inverse of coset theta is equal to theta For all theta lying in the closed interval minus pi by 2 to pi by 2 But here, sin pi by 6 does not lie between minus pi by 2 and pi by 2 So we write sin pi by 6 as s pi by 6 Therefore, the given expression becomes coset inverse of coset of minus pi by 6 Which is equal to coset inverse of coset cos pi by 6 As coset inverse of coset theta will be equal to theta For all theta belonging to the closed interval minus pi by 2 to pi by 2 Where theta cannot be equal to 0 Coset inverse of coset cos theta is equal to theta Therefore, coset inverse of coset of pi by 6 is equal to pi by 6 Therefore, the value of the expression coset inverse of coset of 5 pi by 6 is equal to pi by 6 Which is the required answer Now, we will find the value of the expression cot inverse of cot of 5 pi by 4 We know that cot inverse of cot theta is equal to theta For all theta, line is the open interval 0 to pi But sin pi by 4 does not lie between 0 and pi Pi by 4 pi plus pi by 4 Therefore, the expression cot inverse of cot of 5 pi by 4 becomes cot inverse of cot of pi plus pi by 4 And we know that cot of pi plus theta is equal to cot of theta So, we can write it as cot inverse of cot of pi by 4 Which is equal to pi by 4 Therefore, the value of the expression cot inverse of cot of pi pi by 4 is equal to pi by 4 Which is the required answer Next, we shall find the value of the expression sin of second inverse of x Let second inverse of x be theta Which implies that sin cos theta is equal to x In a triangle A, B, C, if theta is the angle between the lines A, C and C, B We know that second theta is equal to hypotenuse type base That is A, C, F and B, C Which will be equal to x by 1 So, we have the value of the hypotenuse A, C as x And the value of base B, C is given by 1 Then we can find the value of the prependicular A, B by using the Pythagoras theorem By Pythagoras theorem, we have prependicular A, B is given by square root of hypotenuse A, C squared minus base B, C squared Which is equal to square root of x squared minus 1 squared That is, square root of x squared minus 1 So, prependicular A, B is given by square root of x squared minus 1 And we know that sin of theta is given by the prependicular upon hypotenuse That is, A, B upon A, C Which will be equal to square root of x squared minus 1 by x Which implies that theta is given by sin inverse of square root of x squared minus 1 by x Now we have second inverse of x as theta And theta is equal to sin inverse of square root of x squared minus 1 by x Which implies that second inverse of x is equal to sin inverse of square root of x squared minus 1 by x Now substituting the value of second inverse of x in the given expression We get sin of second inverse of x is equal to sin of sin inverse of square root of x squared minus 1 by x Which is equal to square root of x squared minus 1 by x Therefore, the value of the expression sin of second inverse of x is equal to square root of x squared minus 1 by x Which is the required answer Next we shall find the value of the expression cos of tan inverse of 2 by 3 Let tan inverse of 2 by 3 be 5 Which implies that tan of 5 is equal to 2 by 3 In a triangle PQR If psi is the angle between the lines P R and RQ And we know that tan of angle 5 is given by the perpendicular upon base That is PQ upon RQ Which is equal to 2 by 3 So we have the perpendicular PQ as 2 and base RQ as 3 Then we can find the value of the hypotenuse P R by using the Pythagoras theorem By Pythagoras theorem we have hypotenuse P R is equal to square root of the perpendicular PQ squared plus base RQ squared That is square root of 2 squared plus 3 squared Which is equal to square root of 4 plus 9 That is square root of 13 Therefore, hypotenuse P R is equal to square root of 13 Now we know that cos of 5 is given by base upon hypotenuse That is RQ upon R P Which is equal to 3 upon square root of 13 Which implies that phi is equal to cos of 5 3 by square root of 13 Now we have tan of 5 2 by 3 as 5 and phi is equal to cos of 3 by square root of 13 Which implies that tan inverse of 2 by 3 is equal to cos inverse of 3 by square root of 13 Now substituting the value of tan inverse of 2 by 3 in the given expression we get cos of tan inverse of 2 by 3 is equal to cos of cos inverse of 3 by square root of 13 which is equal to 3 by square root of 13 Therefore the value of the expression cos of tan inverse of 2 by 3 is equal to 3 by square root of 13 which is the required answer This completes our session Hope you enjoyed this session