Choose the correct option and justify your choice : 2tan 30°/1+tan230° = ? (A) sin 60° (B) cos 60° (C) tan 60° (D) sin 30°

Answer: (A) sin 60°

Trigonometry Values Table

Trigonometry values for angles 0°, 30°, 45°, 60°and 90°, with respect to sin, cos, tan, cot, sec, cosec functions, taking an example of the right-angle triangle.

Angle

00

300 450 600

900

Sin θ 0 1/2 1/√2 √3/2 1
Cos θ 1 √3/2 1/√2 1/2 0
Tan θ 0 1/√3 1 √3
Cot θ √3 1 1/√3 0
Sec θ 1 2/√3 √2 2
Cosec θ 2 √2 2/√3 1

Substitute the of tan 30° in the given equation

tan 30° = 1/√3

\(\frac{2tan\, 30}{1+ tan^{2}30} = \frac{2\times \frac{1}{\sqrt{3}}}{1+\frac{1}{\sqrt{3}}^{2}}\) \(\frac{2tan\, 30}{1+ tan^{2}30} = \frac{2\sqrt{3}}{1+\frac{1}{3}}\) \(\frac{2tan\, 30}{1+ tan^{2}30} = \frac{2\sqrt{3}}{\frac{4}{3}}\) \(\frac{2tan\, 30}{1+ tan^{2}30} =\frac{6}{4\sqrt{3}}\) \(\frac{2tan\, 30}{1+ tan^{2}30} =\frac{2}{\sqrt{3}}\times \frac{3}{4}\) \(\frac{2tan\, 30}{1+ tan^{2}30} =\frac{2}{\sqrt{3}}\times \frac{\sqrt{3}\sqrt{3}}{4}\) \(\frac{2tan\, 30}{1+ tan^{2}30} =\frac{\sqrt{3}}{2}\)

sin 60° = √3/2

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