Question

How do you express `sin(sin^(-1)(x)+cos^(-1)(y))` without trigonometric functions?

Answer 1

`sin(sin^(-1)(x)+cos^(-1)(y)) = xy+sqrt(1-y^2)sqrt(1-x^2)`

Explanation:

Let `alpha = sin^(-1)(x)` and `beta = cos^(-1)(y)`

Then:

`-pi/2 <= alpha <= pi/2" "` so `" "cos alpha >= 0`

`0 <= beta <= pi" "` so `" "sin beta >= 0`

By Pythagoras:

`cos^2 theta + sin^2 theta = 1`

Hence we have:

`sin(alpha) = sin(sin^(-1)(x)) = x`

`cos(alpha) = sqrt(1-sin^2(alpha)) = sqrt(1-x^2)`

`cos(beta) = cos(cos^(-1)(y)) = y`

`sin(beta) = sqrt(1-cos^2(beta)) = sqrt(1-y^2)`

Noting that we can use the non-negative square root in both these cases from our prior observation that `cos alpha >= 0` and `sin beta >= 0`.

Then using the sum formula for `sin` we find:

`sin(sin^(-1)(x)+cos^(-1)(y)) = sin(alpha+beta)`

`color(white)(sin(sin^(-1)(x)+cos^(-1)(y))) = sin(alpha)cos(beta)+sin(beta)cos(alpha)`

`color(white)(sin(sin^(-1)(x)+cos^(-1)(y))) = xy+sqrt(1-y^2)sqrt(1-x^2)`