Question

verify that the equation is an identity.
\\( \cos ^ { 2 } \theta ( \tan ^ { 2 } \theta + 1 ) = 1 \\)
to verify the identity, start with the more complicated side and transform it to look like the other side. choose t
\\( \

$$\begin{aligned} \\cos ^ { 2 } \\theta ( \\tan ^ { 2 } \\theta + 1 ) & = \\cos ^ { 2 } \\theta ( \\square ) \\\\ & = \\cos ^ { 2 } \\theta ( \\square ) \\\\ & = 1 \\end{aligned}$$

\\)

Explanation:

Step1: Use the Pythagorean identity

We know the Pythagorean identity \(1 + \tan^{2}\theta=\sec^{2}\theta\). So, \(\cos^{2}\theta(\tan^{2}\theta + 1)=\cos^{2}\theta(\sec^{2}\theta)\)

Step2: Use the reciprocal identity

Since \(\sec\theta=\frac{1}{\cos\theta}\), then \(\sec^{2}\theta=\frac{1}{\cos^{2}\theta}\). So, \(\cos^{2}\theta(\sec^{2}\theta)=\cos^{2}\theta\times\frac{1}{\cos^{2}\theta}\)

Step3: Simplify the expression

\(\cos^{2}\theta\times\frac{1}{\cos^{2}\theta}=1\)

Answer:

The blanks should be filled with \(\sec^{2}\theta\), \(\frac{1}{\cos^{2}\theta}\) in order.