Questions: Simplify the expression by using a double-angle formula. (2 tan 5 theta)/(1-tan^2 5 theta)

Solution Steps

We start with the expression: \[ \frac{2 \tan(5\theta)}{1 - \tan^2(5\theta)} \]

This expression can be recognized as the double-angle formula for tangent, which states: \[ \tan(2x) = \frac{2 \tan(x)}{1 - \tan^2(x)} \] In our case, we can let \( x = 5\theta \).

By applying the double-angle formula, we can simplify the expression: \[ \frac{2 \tan(5\theta)}{1 - \tan^2(5\theta)} = \tan(2 \cdot 5\theta) = \tan(10\theta) \]

Final Answer