Question

question 4
one possible way to simplify \\(\cos^2 x + \cos^2 x \cdot \tan^2 x\\) is shown.
choose the correct substitution in each of the steps below.
simplify: \\(\cos^2 x + \cos^2 x \cdot \tan^2 x\\)
step 1: \\(\cos^2 x \left(a + \tan^2 x\
ight)\\) where \\(a = \\) 1
step 2: \\(\cos^2 x \left(b\
ight)\\) where \\(b = \\) \\(\sec^2 x\\)
step 3: \\(\cos^2 x \cdot \frac{1}{c}\\) where \\(c = \\) \\(\cos^2 x\\)
step 4: \\(d\\) where \\(d = \\) 1

Explanation:

Step 1: Factor out \(\boldsymbol{\cos^2 x}\)

In the expression \(\cos^2 x + \cos^2 x \cdot \tan^2 x\), we can factor out \(\cos^2 x\) from both terms. Factoring out a common factor means we divide each term by the common factor and write it outside a set of parentheses. So, \(\cos^2 x \div \cos^2 x = 1\) and \(\cos^2 x \cdot \tan^2 x \div \cos^2 x=\tan^2 x\). Thus, the expression becomes \(\cos^2 x(1 + \tan^2 x)\), so \(A = 1\).

Step 2: Use the Pythagorean Identity for Tangent and Secant

We know the Pythagorean identity \(1+\tan^2 x=\sec^2 x\). So substituting \(1 + \tan^2 x\) with \(\sec^2 x\) in the expression \(\cos^2 x(1 + \tan^2 x)\), we get \(\cos^2 x(\sec^2 x)\), so \(B=\sec^2 x\).

Step 3: Recall the Definition of Secant

The secant function is defined as \(\sec x=\frac{1}{\cos x}\), so \(\sec^2 x = \frac{1}{\cos^2 x}\). In the expression \(\cos^2 x(\sec^2 x)\), we can rewrite \(\sec^2 x\) as \(\frac{1}{\cos^2 x}\), so the expression becomes \(\cos^2 x\cdot\frac{1}{\cos^2 x}\), which means \(C = \cos^2 x\).

Step 4: Simplify the Expression

When we multiply \(\cos^2 x\) by \(\frac{1}{\cos^2 x}\), the \(\cos^2 x\) terms cancel out. \(\cos^2 x\cdot\frac{1}{\cos^2 x}=\frac{\cos^2 x}{\cos^2 x}=1\), so \(D = 1\).

Answer:

• \(A = 1\)
• \(B=\sec^2 x\)
• \(C = \cos^2 x\)
• \(D = 1\)