Question

1. factor to write each in a simpler form.

a) \\( \sec x \sin^2 x - \sec x \\)
b) \\( \sin^4 \theta - \cos^4 \theta \\)

Explanation:

Part (a)

Step1: Factor out \(\sec x\)

Notice that both terms in \(\sec x \sin^2 x - \sec x\) have a common factor of \(\sec x\). So we factor it out:
\(\sec x (\sin^2 x - 1)\)

Step2: Use Pythagorean identity

Recall the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\), which can be rearranged to \(\sin^2 x - 1 = -\cos^2 x\). Substitute this into the expression:
\(\sec x (-\cos^2 x)\)

Step3: Simplify \(\sec x\)

Since \(\sec x=\frac{1}{\cos x}\), substitute that in:
\(\frac{1}{\cos x} \cdot (-\cos^2 x)\)

Step4: Multiply the terms

Multiply \(\frac{1}{\cos x}\) and \(-\cos^2 x\):
\(-\cos x\)

Part (b)

Step1: Recognize difference of squares

The expression \(\sin^4 \theta - \cos^4 \theta\) is a difference of squares, since \(\sin^4 \theta = (\sin^2 \theta)^2\) and \(\cos^4 \theta = (\cos^2 \theta)^2\). The formula for difference of squares is \(a^2 - b^2=(a - b)(a + b)\). So we factor it as:
\((\sin^2 \theta - \cos^2 \theta)(\sin^2 \theta + \cos^2 \theta)\)

Step2: Use Pythagorean identity

Recall that \(\sin^2 \theta + \cos^2 \theta = 1\). Substitute this into the expression:
\((\sin^2 \theta - \cos^2 \theta)(1)\)

Step3: Simplify (optional: use double - angle identity, but the expression \(\sin^2 \theta - \cos^2 \theta\) can also be left as is or we can use the double - angle identity for cosine \(\cos(2\theta)=\cos^2 \theta-\sin^2 \theta\), so \(\sin^2 \theta - \cos^2 \theta=-\cos(2\theta)\))

If we use the double - angle identity, we get \(-\cos(2\theta)\). If we just simplify the factored form, we have \(\sin^2 \theta - \cos^2 \theta\) (or \(-\cos(2\theta)\))

Part (a) Answer: \(-\cos x\)

Part (b) Answer: \(\sin^2 \theta - \cos^2 \theta\) (or \(-\cos(2\theta)\))

Answer:

Step1: Recognize difference of squares

The expression \(\sin^4 \theta - \cos^4 \theta\) is a difference of squares, since \(\sin^4 \theta = (\sin^2 \theta)^2\) and \(\cos^4 \theta = (\cos^2 \theta)^2\). The formula for difference of squares is \(a^2 - b^2=(a - b)(a + b)\). So we factor it as:
\((\sin^2 \theta - \cos^2 \theta)(\sin^2 \theta + \cos^2 \theta)\)

Step2: Use Pythagorean identity

Recall that \(\sin^2 \theta + \cos^2 \theta = 1\). Substitute this into the expression:
\((\sin^2 \theta - \cos^2 \theta)(1)\)

Step3: Simplify (optional: use double - angle identity, but the expression \(\sin^2 \theta - \cos^2 \theta\) can also be left as is or we can use the double - angle identity for cosine \(\cos(2\theta)=\cos^2 \theta-\sin^2 \theta\), so \(\sin^2 \theta - \cos^2 \theta=-\cos(2\theta)\))

If we use the double - angle identity, we get \(-\cos(2\theta)\). If we just simplify the factored form, we have \(\sin^2 \theta - \cos^2 \theta\) (or \(-\cos(2\theta)\))

Part (a) Answer: \(-\cos x\)

Part (b) Answer: \(\sin^2 \theta - \cos^2 \theta\) (or \(-\cos(2\theta)\))