Find the exact value:

Question

sec(2sin^-1(3/4))?

Answer 1

sec(2sin^{- 1} [3/4].

Let, sin^{- 1} (3/4) = t, then, sin t = 3/4.

sec(2sin^{- 1} [3/4] = sec(2t)

Using reciprocal identity : sec x = 1/cos x.

sec(2t) = 1/cos(2t).

Double angle formula : cos(2x) = 1 - 2sin² x.

1/cos(2t) = 1/(1 - 2sin² t)

= 1/(1 - 2(3/4)²)

= 1/(1 - [2*9/16])

= 1/(1 - 9/8)

= 8/(8 - 9)

= - 8.

Therefore, the exact value of sec(2sin^{- 1} [3/4] is - 8.