Question

question
considering only the values of ( \theta ) for which ( \frac{\tan^2 \theta + 1}{sec^3 \theta} ) is defined, which of the following expressions is equivalent to ( \frac{\tan^2 \theta + 1}{sec^3 \theta} )?
select the correct answer below:
( \bigcirc \tan^2 \theta )
( \bigcirc cot^2 \theta )
( \bigcirc sec \theta )
( \bigcirc cos \theta )

Explanation:

Step1: Recall trigonometric identity

We know the Pythagorean identity $\tan^{2}\theta + 1=\sec^{2}\theta$.

Step2: Substitute the identity into the expression

Substitute $\tan^{2}\theta + 1$ with $\sec^{2}\theta$ in the given expression $\frac{\tan^{2}\theta + 1}{\sec^{3}\theta}$. So we get $\frac{\sec^{2}\theta}{\sec^{3}\theta}$.

Step3: Simplify the fraction

Using the rule of exponents for division of like bases $a^{m}\div a^{n}=a^{m - n}$, here $a = \sec\theta$, $m = 2$, $n = 3$. So $\frac{\sec^{2}\theta}{\sec^{3}\theta}=\sec^{2 - 3}\theta=\sec^{- 1}\theta$.

Step4: Recall the reciprocal identity

We know that $\sec\theta=\frac{1}{\cos\theta}$, so $\sec^{-1}\theta=\cos\theta$.

Answer:

$\cos\theta$