Question

Show that each expression can be simplified to cos \(2 x\) a) \(\cos ^{4} x-\sin ^{4} x\) b) \(\frac{\csc ^{2} x-2}{\csc ^{2} x}\)

Short answer

\(\cos ^{4} x-\sin ^{4} x = \cos 2x\), \(\frac{\csc ^{2} x-2}{\csc ^{2} x} = \cos 2x\)

Step-by-step solution

Simplify \(\backslash\text{cos} ^{4} x-\backslash\text{sin} ^{4} x\)

First, recognize that \(\backslash\text{cos} ^{4} x-\backslash\text{sin} ^{4} x\) is a difference of squares. This can be factored as: \(\backslashtext{cos} ^{4} x - \backslashtext{sin} ^{4} x = (\backslashtext{cos} ^{2} x)^2 - (\backslashtext{sin} ^{2} x)^2 = (\backslashtext{cos} ^{2} x - \backslashtext{sin} ^{2} x)(\backslashtext{cos} ^{2} x + \backslashtext{sin} ^{2} x)\).

Apply Pythagorean Identity

Use the Pythagorean identity \(\backslashtext{cos} ^{2} x + \backslashtext{sin} ^{2} x = 1\) to simplify one of the factors: \(\backslashtext{cos} ^{2} x + \backslashtext{sin} ^{2} x = 1\). So, the expression now becomes \((\backslashtext{cos} ^{2} x - \backslashtext{sin} ^{2} x)(1) = \backslashtext{cos} ^{2} x - \backslashtext{sin} ^{2} x\).

Use Double Angle Identity for Cosine

Recognize that \(\backslashtext{cos} ^{2} x - \backslashtext{sin} ^{2} x\) is the double angle identity for cosine. Thus, \(\backslashtext{cos} ^{2} x - \backslashtext{sin} ^{2} x = \backslashtext{cos} 2x\).

Simplify \(\frac{\backslashtext{csc} ^{2} x-2}{\backslashtext{csc} ^{2} x}\)

Rewrite \(\backslashtext{csc} ^{2} x\) in terms of sine: \(\backslashtext{csc} ^{2} x = \frac{1}{\backslashtext{sin}^2 x}\). The expression becomes \(\frac{\frac{1}{\backslashtext{sin} ^{2} x} - 2}{\frac{1}{\backslashtext{sin} ^{2} x}} = \frac{1 - 2 \backslashtext{sin} ^{2} x}{1} = 1 - 2 \backslashtext{sin} ^{2} x\).

Use Pythagorean Identity Again

Recall the identity \(\backslashtext{sin} ^{2} x = 1 - \backslashtext{cos} ^{2} x\). Substitute this into the expression to get: \(1 - 2(1 - \backslashtext{cos} ^{2} x) = 1 - 2 + 2 \backslashtext{cos} ^{2} x = 2 \backslashtext{cos} ^{2} x - 1\).

Use Another Double Angle Identity

Recognize that \(2 \backslashtext{cos} ^{2} x - 1 = \backslashtext{cos} 2x\). Therefore, the expression simplifies to \(\backslashtext{cos} 2x\).

Key concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable. They are essential tools for simplifying complex trigonometric expressions.
Some of the most important trigonometric identities include:

• Pythagorean identities, such as \(\text{sin}^2 x + \text{cos}^2 x = 1\)
• Reciprocal identities, like \(\text{csc} x = \frac{1}{\text{sin} x}\)
• Double angle identities, such as \(\text{cos}(2x) = \text{cos}^2 x - \text{sin}^2 x\)

Using these identities, you can transform and simplify expressions to make solving problems easier. The given exercise involves recognizing and applying these identities.

Pythagorean Identity

The Pythagorean identity is one of the foundational identities in trigonometry. It states that:
\[ \text{sin}^2 x + \text{cos}^2 x = 1 \]
This identity is derived from the Pythagorean theorem applied to a right triangle. It's useful in simplifying trigonometric expressions because it allows us to express \(\text{sin}^2 x\) in terms of \(\text{cos}^2 x\) and vice versa.
For example, in the exercise, we utilized:
\[ \text{cos}^2 x + \text{sin}^2 x = 1 \]
to simplify \( \text{cos}^4 x - \text{sin}^4 x \). By factoring the difference of squares and then applying the Pythagorean identity, we simplified the expression to \( \text{cos}^2 x - \text{sin}^2 x\), which is the cosine double angle identity.

Cosecant Identity

The cosecant function is the reciprocal of the sine function. It is defined as:
\[ \text{csc} x = \frac{1}{\text{sin} x} \]
This identity is especially useful when dealing with expressions that involve reciprocal trigonometric functions.
In the exercise, step 4 involved expressing \( \text{csc}^2 x \) in terms of \(\text{sin}^2 x\):
\[ \text{csc}^2 x = \frac{1}{\text{sin}^2 x} \]
Rewriting \(\text{csc}^2 x\) helped us simplify the fraction, leading to the expression \(1 - 2 \text{sin}^2 x\). We then utilized the Pythagorean identity again to further simplify and recognize this as another form of the cosine double angle identity.