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

Question

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

Quizplus · Accepted Answer

Given $x = 2 \tan(\theta)$, we substitute $x$ into the expression $\sqrt{4 + x^2}$ to get $\sqrt{4 + (2\tan(\theta))^2}$. Simplifying inside the square root gives $\sqrt{4 + 4\tan^2(\theta)}$, which simplifies further to $\sqrt{4(1 + \tan^2(\theta))}$. Using the trigonometric identity $1 + \tan^2(\theta) = \sec^2(\theta)$, we get $\sqrt{4\sec^2(\theta)}$, which simplifies to $2\sec(\theta)$.

If X = 2 Tan θ\theta θ , Use Trigonometric Substitution to Write

If X = 2 Tan $\theta$ , Use Trigonometric Substitution to Write