Prove that sin [ 2 tan inverse 3/5-sin inverse 7/25]=304/425 - Maths - Inverse Trigonometric Functions

Question

Prove that sin [ 2 tan inverse 3/5-sin inverse 7/25]=304/425

Akhil Goyal · Accepted Answer

To Prove:sin2tan-135-sin-1725=304425  ---(1)We know the identity:cos-1x = 2tan-11-x21+x ---(2)Let 1-x21+x=ySquaring and adding 1 on both sides:y2+1=21+x⇒1-y21+y2=xsubstituting in (2):cos-11-y21+y2 = 2tan-1ySubstituting y=352tan-135 = cos-1817After substituting this in (1), we need to prove:sincos-1817-sin-1725=304425Let A =cos-1817 , and B =sin-1725sinA-B = sin A cos B - cos A sin B = 1517×2425-817×725=304425Hence Proved