Find the exact value of each expression

Question

Find the exact value of each expression

a) sin (2sin^(-1) 1/(sqrt2))

b) cot [2 cos^(-1) -3/5]

Answer 1

a) sin(2sin^-1(1/√2)) = sin(2·π/4 = sin(π/2) = 1

b) cot[2cos^-1(-3/5)]; Let cos^-1(-3/5) = α; then cosα = - 3/5 and α in Quadrant II.

cot(2α) = cos2α/sin2α; cos2α = 2cos²α -1 = 2(-3/5)² - 1 = - 7/25;

sin2α = 2sinαcosα = 2(- 3/5)√(1 - 9/25) = - 6/5·4/5 = - 24/25;

cot2α = (- 7/25)(- 25/24) = 7/24

Answer: 7/24