 Hello and welcome to the session. In this session we are going to discuss the following question which says that Find the values of the following. 1. Second verse of square root of 2 plus 2 cos inverse of minus 1 by square root of 3 minus cos inverse of minus 2 by square root of 3 2 tan inverse of 1 plus 2 sin inverse of square root of 3 by 2 minus cos inverse of 1 by 2 for the function y equal to sin inverse of x The principal value of y is less than equal to pi by 2 and is greater than equal to minus pi by 2 for the function y equal to cos inverse of x The principal value of y is less than equal to pi and is greater than equal to 0 For the function y equal to tan inverse of x the principal value of y lies in between minus pi by 2 to pi by 2 For the function y equal to cos inverse of x the principal value of y lies in between 0 to pi For the function y equal to second inverse of x the principal value of y is less than equal to pi and is greater than 0 and the value of y is not equal to pi by 2 For the function y equal to cos inverse of x the principal value of y is less than equal to pi by 2 and is greater than equal to minus pi by 2 and the value of y cannot be 0 With this key idea we shall proceed with the solution We have the expression second inverse of square root of 2 plus 2 cot inverse of minus 1 upon square root of 3 minus cos inverse of minus 2 upon square root of 3 First of all we shall find the value of second inverse of square root of 2 let x1 be second inverse of square root of 2 We know that for the function y equal to second inverse of x the principal value of y is less than equal to pi and is greater than 0 Also y cannot be equal to pi by 2 So here the principal value of x1 is less than equal to pi and is greater than 0 x1 is equal to second inverse of square root of 2 implies that second of x1 is equal to square root of 2 Which can also be written as second of x1 is equal to second pi by 4 as second pi by 4 is equal to square root of 2 Which implies that x1 is equal to pi by 4 Therefore the value of second inverse of square root of 2 is equal to pi by 4 Now let x2 be equal to cot inverse of minus 1 upon square root of 3 for the function y equal to cot inverse of x the principal value of y lies in between 0 to pi So here the principal value of x2 lies in between 0 to pi x2 is equal to cot inverse of minus 1 upon square root of 3 implies that cot of x2 is equal to minus 1 upon square root of 3 Which can also be written as cot of x2 is equal to minus of cot of pi by 3 On we can write cot of x2 is equal to cot of minus pi by 3 Which implies that x2 is equal to minus pi by 3 Therefore the value of cot inverse of minus 1 upon square root of 3 is equal to minus of pi by 3 Now again let x3 be equal to cot inverse of minus 2 upon square root of 3 For the function y equal to cot inverse of x the principal value of y is less than equal to pi by 2 and is greater than equal to minus pi by 2 Where y is not equal to 0 So here the principal value of x3 is less than equal to pi by 2 and is greater than equal to minus pi by 2 x3 is equal to cot of minus 2 upon square root of 3 implies that cot of x3 is equal to minus 2 upon square root of 3 Which can be written as cot of x3 is equal to minus of cot of pi by 3 Or we can write cot of x3 is equal to cot of minus pi by 3 Which implies that x3 is equal to minus pi by 3 Therefore cot inverse of minus 2 upon square root of 3 is equal to minus pi by 3 Now we have got second inverse of square root of 2 is equal to pi by 4 cot inverse of minus 1 upon square root of 3 is equal to minus pi by 3 and cot second inverse of minus 2 upon square root of 3 is equal to minus pi by 3 Now substituting all these values in the given expression we get Second inverse of square root of 2 plus 2 cot inverse of minus 1 upon square root of 3 minus cot second inverse of minus 2 upon square root of 3 is equal to pi by 4 plus 2 into minus pi by 3 minus of minus pi by 3 which is equal to pi by 4 minus 2 pi by 3 plus pi by 3 that is pi by 4 minus pi by 3 which is equal to minus pi by 12 Therefore the value of the expression second inverse of square root of 2 plus 2 cot inverse of minus 1 upon square root of 3 minus cot second inverse of minus 2 upon square root of 3 is given by minus pi by 12 which is the required answer Next we shall find the value of the expression tan inverse of 1 plus 2 sin inverse of square root of 3 by 2 minus cos inverse of 1 by 2 We have the expression tan inverse of 1 plus 2 sin inverse of square root of 3 by 2 minus cos inverse of 1 by 2 First we shall find the value of tan inverse of 1 Let X be tan inverse of 1 SUGIT function phi equal to tan inverse of X the principal value of Y lies in between minus pi by 2 to pi by 2 Therefore the principal value of X will lie in between minus pi by 2 to pi by 2 X is equal to tan inverse of 1 implies that tan of X is equal to 1 and we can write tan of X is equal to tan of pi by 4 implies that X is equal to pi by 4 Therefore tan inverse of 1 is equal to pi by 4 Now let Y be equal to sin inverse of square root of 3 by 2 SUGIT function Y equal to sin inverse of X the principal value of Y is less than equal to pi by 2 and is greater than equal to minus pi by 2 So here the principal value of Y is less than equal to pi by 2 and is greater than equal to minus pi by 2 Y is equal to sin inverse of square root of 3 by 2 implies that sin of Y is equal to square root of 3 by 2 which can be written as sin of Y is equal to sin of pi by 3 which implies that Y is equal to pi by 3 Therefore sin inverse of square root of 3 by 2 is equal to pi by 3 Now let Z be equal to cos inverse of 1 by 2 SUGIT function Y equal to sin inverse of X the principal value of Y is less than equal to pi and is greater than equal to 0 Therefore the principal value of Z will be less than equal to pi and greater than equal to 0 Z is equal to cos inverse of 1 by 2 implies that cos of Z is equal to 1 by 2 which can be written as cos of Z is equal to cos of pi by 3 which implies that Z is equal to pi by 3 Therefore cos inverse of 1 by 2 is equal to pi by 3 Now we have sin inverse of 1 is equal to pi by 4 Sin inverse of square root of 3 by 2 is equal to pi by 3 and cos inverse of 1 by 2 is equal to pi by 3 Now substituting all these values in the given expression we get tan inverse of 1 plus 2 sin inverse of square root of 3 by 2 is equal to cos inverse of 1 by 2 is equal to pi by 4 plus 2 pi by 3 minus pi by 3 which is equal to pi by 4 plus pi by 3 that is 7 pi by 12 Therefore the value of the expression tan inverse of 1 square root of 3 by 2 minus cos inverse of 1 by 2 is given by 7 pi by 12 which is the required answer This completes our session Hope you enjoyed this session