cosec[ 2cot ⁻¹(5) + cos ⁻¹( 4/5 ) ] is equal to :

Solution

Maths sir · 3 steps

Turn 2 cot⁻¹ 5 into a single angle with known sine and cosine, then use the sine addition formula.

1. Double the angle. Let θ = cot⁻¹ 5, so tan θ = 1/5. Then tan 2θ = (2/5)/(1 - 1/25) = (2/5)/(24/25) = 5/12. Since θ < π/4, the angle 2θ lies in the first quadrant, so sin 2θ = 5/13 and cos 2θ = 12/13.
2. Second angle. Let φ = cos⁻¹ 4/5. Then cos φ = 4/5 and sin φ = 3/5, from the 3-4-5 triangle.

   sin(2θ + φ) = 5/13·4/5 + 12/13·3/5 = (20 + 36)/65 = 56/65

3. So. cosec(2θ + φ) = 65/56, which is option (b).

The idea to keep. Write every inverse trig term as an angle and read off its sine and cosine from a right triangle. Then the addition formulas do the rest. Check the quadrant before you take square roots.

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