Question

Prove that sin-1(5/13)+cos-1(4/5) =1/2 sin- (3626/4223)

Answer 1

Answered

Prove that : sin^-1(5/13) + cos^-1(4/5) =1/2sin^-1(3696/4225)

2

SEE ANSWERS

abhi178

abhi178

we have to prove sin^-1(5/13) + cos^-1(4/5) = 1/2 sin^-1(3696/4225)

or, 2sin^-1(5/13) + 2cos^-1(4/5) = sin^-1(3696/4225)

let sin^-1(5/13) = x

so, sinx = 5/13 , cosx = 12/13

we know, sin2x = 2sinx.cosx

= 2 × 5/13 × 12/13 = 120/169

and cos2x = 1 - 2sin²x = 1 - 2 × (5/13)²

= (169 - 50)/169 = 119/169

similarly, cos^-1(4/5) = y

cosy = 4/5 , siny = 3/5

sin2y = 2 × 4/5 × 3/5 = 24/25

cos2y = 1 - 2(3/5)² = 7/25

now, sin(2x + 2y) = sin2x.cos2y + cos2x.sin2y

= 120/169 × 7/25 + 119/169 × 24/25

= (840 + 2856)/4225

= 3696/4225

hence, sin(2x + 2y) = 3696/4225

or, 2x + 2y = sin^-1(3696/4225)

now, LHS = 2sin^-1(5/13) + 2cos^-1(4/5)

= 2x + 2y

= sin^-1(3696/4225) = RHS

hence, it is proved that 2sin^-1(5/13) + 2cos^-1(4/5) = sin^-1(3696/4225)

so, it is also proved that sin^-1(5/13) + cos^-1(4/5) = 1/2sin^-1(3696/4225)

plz mark me as a brainlist