using the trigonometric substitution, x = 6 sec θ, given 0 ≤ θ < π/2. simplify the expression below to a…

Explanation:

Step1: Substitute $x = 6\sec\theta$

Substitute $x$ into the expression $\frac{\sqrt{x^{2}-36}}{x}$, we get $\frac{\sqrt{(6\sec\theta)^{2}-36}}{6\sec\theta}$.

Step2: Simplify the numerator

First, expand $(6\sec\theta)^{2}-36 = 36\sec^{2}\theta - 36=36(\sec^{2}\theta - 1)$. Since $\sec^{2}\theta-1=\tan^{2}\theta$, then $\sqrt{36(\sec^{2}\theta - 1)}=\sqrt{36\tan^{2}\theta}=6|\tan\theta|$. Given $0\leq\theta<\frac{\pi}{2}$, $\tan\theta\geq0$, so it is $6\tan\theta$.

Step3: Simplify the whole - expression

The expression becomes $\frac{6\tan\theta}{6\sec\theta}$. Since $\sec\theta=\frac{1}{\cos\theta}$, we have $\frac{6\tan\theta}{6\sec\theta}=\frac{\frac{6\sin\theta}{\cos\theta}}{\frac{6}{\cos\theta}}=\sin\theta$.

Answer:

$\sin\theta$