If the inverse trigonometric functions take principal values then
cos ⁻¹( 3/10 cos ( tan ⁻¹( 4/3 ) ) + 2/5 sin ( tan ⁻¹( 4/3 ) ) ) is equal to :

Solution

Maths sir · 3 steps

Draw the 3-4-5 triangle for tan⁻¹ 4/3 and the inside collapses to one half.

1. Name the angle. Let θ = tan⁻¹ 4/3. Since 4/3 > 0, the principal value puts θ in (0, π/2), so sin θ and cos θ are both positive.
2. Use the triangle. Opposite side 4 and adjacent side 3 give hypotenuse √(3² + 4²) = 5. So cos θ = 3/5 and sin θ = 4/5.

   3/10·3/5 + 2/5·4/5 = 9/50 + 16/50 = 25/50 = 1/2

3. So. The expression becomes cos⁻¹ 1/2. The angle in [0, π] with cosine 1/2 is π/3, which is option (c).

The idea to keep. When you see sin or cos of tan⁻¹ of a number, draw the right triangle, find the hypotenuse and read off the ratios. Check the principal range for signs.

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