Question

tan+1(1/2)+tan-1(1/7)=sin-1(31/25√2)

Answer 1

Answer:

Step-by-step explanation:

2 tan⁻¹ (1/2) + tan⁻¹ (1/7)

∴ tan⁻¹ ( (2 x 1/2)/ ( 1 - (1/2)²) + tan⁻¹ (1/7)

∴ tan⁻¹ ( 1 / ( 1 - 1/4 ) + tan⁻¹ (1/7)

∴ tan⁻¹ ( 4/3 ) + tan⁻¹ (1/7)

∴ tan⁻¹ ( (4/3 + 1/7) / 1 - (4/3 * 1/7))

-- Using Formula of tan⁻¹ A + tan⁻¹ B = tan⁻¹ (A+B/ 1 -AB)

∴ tan⁻¹ ( 31 / 17)

Now, Let us say tan⁻¹ ( 31 / 17) = Ф ... (1)

∴ 31/17 = tan Ф .. So 2 sides of right triangle is 31, 17. Let us find hyp.

Hyp = √(31² + 17²)

Hyp = 25√2

sin Ф = 31 / 25√2

∴ Ф = sin⁻¹ (31 / 25√2) --- (2)

So tan⁻¹ ( 31 / 17) = sin⁻¹ (31 / 25√2)

2 tan⁻¹ (1/2) + tan⁻¹ (1/7) = sin⁻¹ (31 / 25√2)

Hence Proved.