sin^-1 (4/5) - cos^-1 (12/13) = cos^-1 (56/65)?

Question

Verufy the equation without using a calculator.

Answer 1

Verify sin^-1 (4/5) - cos^-1 (12/13) = cos^-1 (56/65).

α = sin^-1 (4/5) = cos^-1 (3/5)

Now we have:

cos^-1 (3/5) - cos^-1 (12/13) = cos^-1 (56/65)

β = cos^-1 (12/13)

cosα = 3/5
sinα = 4/5

cosβ = 12/13
sinβ = 5/13

cos(α - β) = (cosα)(cosβ) + (sinα)(sinβ)

cos(α - β) = (3/5)(12/13) + (4/5)(5/13) = (36 + 20)/(5*13)

cos(α - β) = 56/65

cos^-1 (56/65) = α - β

Therefore

sin^-1 (4/5) - cos^-1 (12/13) = cos^-1 (56/65)