use the figure to evaluate the following function, given that h(x)=tan x. h(2θ) h(2θ)=□ (simplify your…

Answer:

$\frac{2a}{a^{2}-1}$

Explanation:

Step1: Find the value of (a)

Since the point ((a,1)) lies on the circle (x^{2}+y^{2}=5), substitute (x = a) and (y = 1) into the equation: (a^{2}+1^{2}=5), so (a^{2}=4), then (a=\pm2). From the position of the point in the first - quadrant (assumed from the context of the angle (\theta)), (a = 2).

Step2: Recall the double - angle formula for tangent

The double - angle formula for tangent is (\tan(2\theta)=\frac{2\tan\theta}{1 - \tan^{2}\theta}).

Step3: Find (\tan\theta)

We know that for the point ((a,1)) on the terminal side of the angle (\theta), (\tan\theta=\frac{y}{x}=\frac{1}{a}).

Step4: Substitute (\tan\theta) into the double - angle formula

(\tan(2\theta)=\frac{2\times\frac{1}{a}}{1-\left(\frac{1}{a}\right)^{2}}=\frac{\frac{2}{a}}{\frac{a^{2}-1}{a^{2}}}=\frac{2a}{a^{2}-1}).