The value of tan ( 2tan ⁻¹( 3/5 ) + sin ⁻¹( 5/13 ) ) is equal to :

Solution

Maths sir · 4 steps

Turn each inverse function into a plain angle with a known tangent, then use the tan of a sum.

1. Double the first angle. Let θ = tan⁻¹ 3/5, so tan θ = 3/5. Then tan 2θ = (2 × 3/5)/(1 - 9/25) = (6/5)/(16/25) = 15/8.
2. Convert the sine. Let φ = sin⁻¹ 5/13. In a right triangle with opposite side 5 and hypotenuse 13, the adjacent side is 12, so tan φ = 5/12.

   tan(2θ + φ) = (15/8 + 5/12)/(1 - 15/8 × 5/12) = (55/24)/(21/96) = (55 × 4)/21

3. Sense check. Since 3/5 < 1, θ < 45°, and 2θ ≈ 62°. With φ ≈ 23°, the total is about 85°, below 90°. So the tangent must be large and positive.
4. So. The value is 220/21, option (b).

The idea to keep. Rewrite every inverse trig term as an angle with a known tangent, then use tan 2A and tan(A+B). A rough estimate of the total angle tells you the sign of the answer.

Answer from the NTA final answer key. Solution by PrepWiser.