Let and be the roots of the quadratic equation x^2 - 8x + 4 = 0 . Then the value of ^2 ( ⁻¹ ( ) ) + cosec ^2 ( ⁻¹ ( ) ) is equal to

Step-by-step solution

Given that and are the roots of x^2 - 8x + 4 = 0 , we have: + = 8 = 4 The expression to be evaluated is: ^2 ( ⁻¹ ( ) ) + cosec ^2 ( ⁻¹ ( ) ) Using the trigonometric identities ^2 = 1 + ^2 and cosec ^2 = 1 + ^2 , the expression simplifies to: 1 + ( )^2 + 1 + ( )^2 = 2 + ^2 ^2 + ^2 ^2 = 2 ^2 ^2 + ^4 + ^4 ^2 ^2 = ( ^2 + ^2)^2 ( )^2 Now, we find the value of ^2 + ^2 : ^2 + ^2 = ( + )^2 - 2 ^2 + ^2 = (8)^2 - 2(4) = 64 - 8 = 56 Substituting the values back into the simplified expression: (56)^2 (4)^2 = ( 56 4 )^2 = (14)^