 Hello and welcome to the session. In this session we are going to discuss the following question which says that simplify 1, tan of 2 cos inverse of x to tan of 2 sin inverse of x. We know that 2 cos inverse of x is equal to cos inverse of 2x square minus 1 and 2 sin inverse of x is equal to sin inverse of 2x into square root of 1 minus x square. With this key idea let us proceed with the solution. We are given the expression tan of 2 cos inverse of x and we know that 2 cos inverse of x can be written as cos inverse of 2x square minus 1. Therefore tan of 2 cos inverse of x can be written as tan of cos inverse of 2x square minus 1. Let cos inverse of 2x square minus 1 be theta which implies that cos of theta is equal to 2x square minus 1. In a triangle ABC if theta is the angle between the lines AC and CB then cos theta is given by base upon hypotenuse that is BC upon AC which is equal to 2x square minus 1 by 1 and if we are given base BC as 2x square minus 1 and hypotenuse AC as 1 then we can find out the perpendicular AB by using the Pythagoras theorem. By Pythagoras theorem we have perpendicular AB is given by square root of hypotenuse AC square minus base BC square which is equal to square root of 1 square minus 2x square minus 1 by whole square which is equal to square root of 1 minus 4x to the power 4 minus x square plus 1 which is equal to square root of 1 minus 4 into x raise to the power 4 plus 4x square minus 1 which can be written as square root of minus 4x to the power 4 plus 4x square which is equal to square root of 4x square into 1 minus x square that is 2x into square root of 1 minus x square therefore perpendicular AB is equal to 2x into square root of 1 minus x square now tan of angle theta is given by perpendicular upon base that is AB upon BC which is equal to 2x into square root of 1 minus x square upon 2x square minus 1 and we have assumed the value of theta as cos inverse of 2x square minus 1 therefore we can write tan of cos inverse of 2x square minus 1 is equal to 2x into square root of 1 minus x square upon 2x square minus 1 and 2 cos inverse of x is equal to cos inverse of 2x square minus 1 therefore tan of 2 cos inverse of x is equal to 2x into square root of 1 minus x square upon 2x square minus 1 hence we can say that the value of the expression tan of 2 cos inverse of x is equal to 2x into square root of 1 minus x square upon 2x square minus 1 next we shall find the value of tan of 2 sin inverse of x and we know that 2 sin inverse of x is equal to sin inverse of 2x into square root of 1 minus x square therefore we can replace 2 sin inverse of x with sin inverse of 2x into square root of 1 minus x square therefore tan of 2 sin inverse of x is equal to tan of sin inverse of 2x into square root of 1 minus x square let sin inverse of 2x into square root of 1 minus x square d theta which implies that sin of theta is equal to 2x into square root of 1 minus x square in a triangle A, B, C if theta is the angle between the lines A, C and C, B then sin of angle theta is given by the Panjuku-Rappan hypotenuse that is A, B upon A, C which is equal to 2x into square root of 1 minus x square upon 1 if we are given the Panjuku-Rappan A, B as 2x into square root of 1 minus x square and hypotenuse A, C as 1 then we can find the value of the base B, C by using Pythagorean theorem by Pythagorean theorem we have base B, C is given by square root of hypotenuse A, C square minus perpendicular A, B square which is equal to square root of 1 square minus 2x into square root of 1 minus x square d whole square that is square root of 1 minus 4x square into 1 minus x square which is equal to square root of 1 minus 4x square minus 4 into x raised to the power 4 that is square root of 1 minus 4x square plus 4 into x raised to the power 4 therefore the value of the base B, C is equal to square root of 4 into x raised to the power 4 minus 4x square plus 1 and we know that tan upon by theta is given by perpendicular upon base that is A, B upon B, C which is equal to 2x into square root of 1 minus x square upon square root of 4 into x raised to the power 4 minus 4x square plus 1 and we have assumed theta as sin inverse of 2x into square root of 1 minus x square so we can write tan of sin inverse of 2x into square root of 1 minus x square is given by 2x into square root of 1 minus x square upon square root of 4 into x raised to the power 4 minus 4x square plus 1 and we know that sin inverse of 2x into square root of 1 minus x square is equal to 2 sin inverse of x therefore the value of tan of 2 sin inverse of x is equal to 2x into square root of 1 minus x square upon square root of 4 into x raised to the power 4 minus 4x square plus 1 which is the required answer this completes our session hope you enjoyed this session