If $x = 2 \cot \theta$, use trigonometric substitution to write $\sqrt { 4 + x ^ { 2 } }$ as a trigonometric function of $\theta$, where $0

Question

If $x = 2 \cot \theta$, use trigonometric substitution to write $\sqrt { 4 + x ^ { 2 } }$ as a trigonometric function of $\theta$, where $0 < \theta < \pi$.
A) $2 \cos \theta$
B) $2 \csc \theta$
C) $2 \cot \theta$
D) $2 \sec \theta$
E) $2 \sin \theta$

Quizplus · Accepted Answer

Given $x = 2 \cot \theta$, we can rewrite $\sqrt{4 + x^2}$ as $\sqrt{4 + (2 \cot \theta)^2}$. Simplifying inside the square root gives $\sqrt{4 + 4 \cot^2 \theta}$. Recognizing that $1 + \cot^2 \theta = \csc^2 \theta$, we can rewrite the expression inside the square root as $4\csc^2 \theta$, which simplifies to $2\csc \theta$ when taking the square root.

If x=2cot⁡θx = 2 \cot \thetax=2cotθ , Use Trigonometric Substitution to Write

If $x = 2 \cot \theta$ , Use Trigonometric Substitution to Write