Question

Rewrite the expression as single function of an angle, if \[ \frac{2\tan31^\circ}{1-\tan^2 31^\circ}=? \]

• A. \(\tan62^\circ\)
• B. \(\tan31^\circ\)
• C. \(\tan15^\circ\)
• D. \(\tan10^\circ\)

Hint:

Remember the tangent double angle formula: \[ \tan 2A=\frac{2\tan A}{1-\tan^2 A} \]

Answer 1

The Correct Option is A

Concept:
We use the double angle identity for tangent: \[ \tan 2A=\frac{2\tan A}{1-\tan^2 A} \] The given expression is: \[ \frac{2\tan31^\circ}{1-\tan^2 31^\circ} \] This exactly matches the formula: \[ \frac{2\tan A}{1-\tan^2 A} \]

Step 1: Compare with the standard formula.
Standard formula: \[ \tan 2A=\frac{2\tan A}{1-\tan^2 A} \] Given expression: \[ \frac{2\tan31^\circ}{1-\tan^2 31^\circ} \] So: \[ A=31^\circ \]

Step 2: Apply the double angle formula.
\[ \frac{2\tan31^\circ}{1-\tan^2 31^\circ} = \tan(2\times31^\circ) \] \[ =\tan62^\circ \] Hence, the correct answer is: \[ \boxed{(A)\ \tan62^\circ} \]