Question

Prove the following:

`cos^-1(3/5) + cos^-1(4/5) = pi/(2)`

Answer 1

Let `cos^-1(3/5)` = x

∴ cos x = `(3)/(5), "where"  0 < x < pi/(2)`
∴ sin x > 0

Now,

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

= `sqrt(1 - 9/25)`

= `sqrt(16/25)`

= `(4)/(5)`
∴ x = `sin^-1(4/5)`

∴ `cos^-1(3/5) = sin^-1(4/5)`    ...(1)

L.H.S. = `cos^-1(3/5) + cos^-1(4/5)`

= `sin^-1(4/5) + cos^-1(4/5)`   ...[By (1)]

= `pi/(2)                      ...[∵ sin^-1x + cos^-1x = pi/2]`

= R.H.S.