tan ⁻¹( (1 + √3)/(3 + √3) ) + sec ⁻¹( √((8 + 4√3)/(6 + 3√3)) ) is equal to :

Solution

Maths sir · 4 steps

Factor out the √3 and the (2+√3), and both angles turn out to be π/6.

1. First term. Write 3 + √3 = √3(√3 + 1). The factor 1 + √3 cancels, so the fraction is 1/√3 and tan⁻¹ 1/√3 = π/6.
2. Second term. Take out common factors: 8 + 4√3 = 4(2 + √3) and 6 + 3√3 = 3(2 + √3). The ratio is 4/3, and its square root is 2/√3.
3. Read the angle. sec θ = 2/√3 means cos θ = √3/2. In the principal range of sec⁻¹ this gives θ = π/6.

   tan⁻¹ 1/√3 + sec⁻¹ 2/√3 = π/6 + π/6 = π/3

4. So. The expression equals π/3, which is option (b).

The idea to keep. Ugly surds inside an inverse trig function almost always hide a standard value. Look for a common factor to cancel before you try any identity.

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