Inverse Functions Help! (1 Viewer)

[lookoutastroboy]

Hey peeps,

Could anyone please provide worked solutions to the Qs below? Any help is much appreciated.

1. Prove tan inverse (x) + tan inverse (1/x) = Pi/2

2. Prove 2 cos inverse x = cos inverse (2x^2 - 1)

3. Prove cos inverse a + cos inverse b = cos inverse (ab - sqrt(1 - a^2) x sqrt(1-b^2))




Cheers,
l.a.

 

[Deep Blue]

1. Take the tan(LHS) and expand. The tan and tan-inverse cancel giving you a formula in x which simplifies to pi/2

 

[xV1P3R]

1. Differentiate the LHS to get 0. So because it's constant, just sub in 1, to get pi/2 which is true for all x in the domain x>0

 

[hup]

 

[Trebla]

1. Prove tan inverse (x) + tan inverse (1/x) = Pi/2

Let y = tan^-1 x
=> x = tan y
=> 1/x = cot y
=> 1/x = tan(π/2 - y)
=> tan^-1(1/x) = π/2 - y
and the result follows