 Hello and welcome to the session. In this session we are going to discuss the following question which says that Evaluate 1 sec of 2 tan inverse of x to sin 1 by 2 into cos inverse of 7 by 11. We know that 2 tan inverse of x is equal to tan inverse of 2 x upon 1 minus x squared and sin of theta by 2 is equal to square root of 1 minus cos theta by 2. With this key idea we shall proceed with the solution. We need to find the value of the expression that is sec of 2 tan inverse of x. 2 tan inverse of x is given by tan inverse of 2 x upon 1 minus x squared. Therefore we can replace 2 tan inverse of x with tan inverse of 2 x upon 1 minus x squared. So we get sec of tan inverse of 2 x upon 1 minus x squared 2 x upon 1 minus d theta which implies that tan inverse theta is equal to 2 x upon 1 minus x squared if theta is the angle between the lines a p and c b. Then tan of angle theta is given by perpendicular upon base that is a b upon b c which is equal to 2 x upon 1 minus x squared if we are given the value of the perpendicular a b as 2 x and the value of the base c b as 1 minus x squared we can find the value of the hypotenuse a c by using Pythagoras theorem. By Pythagoras theorem we have hypotenuse a c is equal to square root of perpendicular a b squared plus base b c squared that is square root of 2 x squared plus 1 minus x squared v whole squared which is equal to square root of 4 x squared plus 1 plus x raised to the power 4 minus 2 x squared which is equal to square root of x raised to the power 4 plus 2 x squared plus 1 which can be written as square root of x squared plus 1 v whole squared that is x squared plus 1 therefore hypotenuse a c is equal to x squared plus 1. Now we know that sec of angle theta is equal to hypotenuse by base that is a c upon b c which is equal to x squared plus 1 upon 1 minus x squared and we have assumed the value of theta as tan inverse of 2 x upon 1 minus x squared so we can write sec of tan inverse of 2 x upon 1 minus x squared is equal to x squared plus 1 upon 1 minus x squared and we know that tan inverse of 2 x upon 1 minus x squared is equal to 2 tan inverse of x therefore the value of the expression sec of 2 tan inverse of x is equal to x squared plus 1 upon 1 minus x squared which is the required answer. Next we shall find the value of the expression sin of 1 by 2 into cos inverse of 7 by 11 let cos inverse of 7 by 11 be theta which implies that cos of theta is equal to 7 by 11 sin of theta by 2 is equal to square root of 1 minus cos theta whole upon 2 therefore we have sin of theta by 2 is equal to square root of 1 minus cos theta that is 7 by 11 whole upon 2 which is equal to square root of 11 minus 7 whole upon 11 divided by 2 that is square root of 4 by 11 into 2 that is square root of 2 by 11 theta by 2 is equal to square root of 2 by 11 and we have assumed theta as cos inverse of 7 by 11 which implies that sin of 1 by 2 into cos inverse of 7 by 11 is equal to square root of 2 by 11 therefore the value of the expression sin of 1 by 2 into cos inverse of 7 by 11 is equal to square root of 2 by 11 which is the required answer this completes our session hope you enjoyed this session