need help with a confusing trig question.

Question

Write an expression equivalent to csc((2cot^-1)x), yet that does not use trigonometric functions. Assume the variable represents a positive value.

Answer 1

csc[2cot^-1x]   Let u = cot^-1x   So, cotu = x = x/1

 

Draw a right triangle with an acute angle u, side adjacent to u = x, and side opposite u = 1.  By the Pythagorean Theorem, the hypotenuse is √(x²+1).  

 

So, csc[2cot^-1x] = csc(2u) = 1/sin(2u)

 

                                               = 1/(2cosusinu)

 

                                               = 1/[2(x/√(x²+1))(1/√(x²+1))

 

                                               = (x²+1)/(2x)