To prove the identity [tex]\(\cos^2 x\left(1 + \tan^2 x\right) = 1\)[/tex], we need to fill in the rules that justify each transformation step. Here's a detailed step-by-step solution:
1. Starting Expression:
[tex]\[
\cos^2 x(1 + \tan^2 x)
\][/tex]
2. Step 1 (Using a Trigonometric Identity):
[tex]\[
\cos^2 x(1 + \tan^2 x) = \cos^2 x(\sec^2 x)
\][/tex]
Rule: This step uses the trigonometric identity [tex]\(1 + \tan^2 x = \sec^2 x\)[/tex].
3. Step 2 (Definition of Secant):
[tex]\[
\cos^2 x(\sec^2 x) = \cos^2 x\left(\frac{1}{\cos^2 x}\right)
\][/tex]
Rule: This step uses the definition of secant, [tex]\(\sec x = \frac{1}{\cos x}\)[/tex], thus [tex]\(\sec^2 x = \frac{1}{\cos^2 x}\)[/tex].
4. Step 3 (Simplification):
[tex]\[
\cos^2 x\left(\frac{1}{\cos^2 x}\right) = 1
\][/tex]
Rule: This step simplifies [tex]\(\cos^2 x \cdot \frac{1}{\cos^2 x} = 1\)[/tex] since the product of a term and its reciprocal is 1.
Combining these steps, we have:
[tex]\[
\cos^2 x(1 + \tan^2 x) = \cos^2 x(\sec^2 x) \quad \text{(Trigonometric Identity)}
\][/tex]
[tex]\[
= \cos^2 x\left(\frac{1}{\cos^2 x}\right) \quad \text{(Definition of Secant)}
\][/tex]
[tex]\[
= 1 \quad \text{(Simplification)}
\][/tex]
So, the rules justifying each step are:
\begin{tabular}{ll}
[tex]$\cos ^2 x\left(1+\tan ^2 x\right)$[/tex] & Initial Expression \\
[tex]$=\cos ^2 x\left(\sec ^2 x\right)$[/tex] & Rule 1: Trigonometric Identity \\
[tex]$=\cos ^2 x\left(\frac{1}{\cos ^2 x}\right)$[/tex] & Rule 2: Definition of Secant \\
[tex]$=1$[/tex] & Rule 3: Simplification of Multiplication \\
\end{tabular}
Thus, the proof is complete.
