cos (sin ⁻¹ 3/5+sin ⁻¹ 5/13+sin ⁻¹ 33/65) is equal to:

Solution

Maths sir · 4 steps

Merge the first two angles into one, and the third turns out to be its complement.

1. Name the angles. Let A = sin⁻¹ 3/5 and B = sin⁻¹ 5/13. Both are acute, so cos A = 4/5 and cos B = 12/13.

   cos(A+B) = 4/5·12/13 - 3/5·5/13 = (48 - 15)/65 = 33/65

2. Merge into one. Each of A and B is below π/2, so A + B lies between 0 and π. On that interval, cos(A+B) = 33/65 fixes the angle: A + B = cos⁻¹ 33/65.
3. Use the complement. Since sin⁻¹ x + cos⁻¹ x = π/2, the full sum is cos⁻¹ 33/65 + sin⁻¹ 33/65 = π/2.
4. So. The expression is cos π/2 = 0, which is option (d).

The idea to keep. When three inverse-sine terms are added, combine two of them first. If the result matches the third number, look for sin⁻¹ x + cos⁻¹ x = π/2.

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