Problem 13 - Algebra and Trigonometry

Question

9–14 Use the formulas for lowering powers to rewrite the expression in terms of the first power of cosine, as in Example 4. $$\cos ^{4} x \sin ^{4} x$$

Step-by-Step Solution

Answer

Use identities to express \(\sin^4 x\) and \(\cos^4 x\), then simplify the product.

Step 1: Understand the Task

We need to express the given expression \( \cos^4 x \sin^4 x \) in terms of the first power of cosine.

Step 2: Use Trigonometric Identity for Sine and Cosine

Recall the identity \( \sin^2 x = 1 - \cos^2 x \). This will help in expressing the power of sine in terms of cosine, which is needed for this exercise.

Step 3: Lowering Powers Formula for Sine

Use the formula \( \sin^2 x = \frac{1}{2}(1 - \cos(2x)) \). Substitute \( \sin^4 x = (\sin^2 x)^2 \) with this identity:

Step 4: Substitute and Simplify \( \sin^4 x \)

Substitute \( \sin^2 x \) in \( \sin^4 x = (\sin^2 x)^2 \):\[\sin^4 x = \left(\frac{1}{2}(1 - \cos(2x))\right)^2 = \frac{1}{4}(1 - \cos(2x))^2.\]

Step 5: Expand \( (1 - \cos(2x))^2 \)

Expand the expression for \( \sin^4 x \):\[(1 - \cos(2x))^2 = 1 - 2\cos(2x) + \cos^2(2x)\]

Step 6: Simplify \( \cos^2(2x) \)

Use the identity \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \) with \( \theta = 2x \) to convert \( \cos^2(2x) \) into first power:\[ \cos^2(2x) = \frac{1 + \cos(4x)}{2}. \]

Step 7: Combine and Simplify Expressions

Substitute back and simplify to express \( \sin^4 x \):\[ \sin^4 x = \frac{1}{4}(1 - 2\cos(2x) + \frac{1 + \cos(4x)}{2}) \]\[ = \frac{1}{4}(1 - 2\cos(2x) + \frac{1}{2} + \frac{1}{2}\cos(4x)) \]\[ = \frac{3}{8} - \frac{1}{2} \cos(2x) + \frac{1}{8} \cos(4x). \]

Step 8: Express \( \cos^4 x \) in terms of \( \cos(2x) \)

Similarly, express: \[\cos^4 x = (\cos^2 x)^2 = \left(\frac{1 + \cos(2x)}{2}\right)^2 \]\[ = \frac{1}{4}(1 + 2\cos(2x) + \cos^2(2x)) \]\[ = \frac{1}{4}(1 + 2\cos(2x) + \frac{1 + \cos(4x)}{2}) \]\[ = \frac{1}{4} \left( \frac{3}{2} + 2\cos(2x) + \frac{1}{2}\cos(4x) \right) \]\[ = \frac{3}{8} + \frac{1}{2}\cos(2x) + \frac{1}{8}\cos(4x) \].

Step 9: Combine all transformed terms

Now, combine \( \cos^4 x \sin^4 x = (\cos^4 x)(\sin^4 x) \) into one simplified expression:\[\cos^4 x \sin^4 x = \left(\frac{3}{8} + \frac{1}{2}\cos(2x) + \frac{1}{8}\cos(4x)\right) \left(\frac{3}{8} - \frac{1}{2}\cos(2x) + \frac{1}{8}\cos(4x)\right)\]Use distribution to find the final expression.

Step 10: Final Simplification

Perform the multiplication and simplify:\[\cos^4 x \sin^4 x = \left(\frac{3}{8}\right)^2 - \left(\frac{3}{8}\right)\left(\frac{1}{2}\cos(2x)\right) + \left(\frac{3}{8}\right)\left(\frac{1}{8}\cos(4x)\right) - \left(\frac{1}{2}\cos(2x)\right)^2 + \left(\frac{1}{2}\cos(2x)\right)\left(\frac{1}{8}\cos(4x)\right) - \left(\frac{1}{8}\cos(4x)\right)^2\]Calculate each term, combine similar terms and simplify.

Key Concepts

Power Reduction Formula

The power reduction formula is a useful tool in trigonometry for simplifying expressions like \( \cos^4 x \sin^4 x \). It helps to reduce the powers of trigonometric functions to lower degrees, often down to the first power, which is more manageable. The formula for the sine function is:
\[\sin^2 x = \frac{1}{2}(1 - \cos(2x))\]This identity allows us to express \( \sin^4 x = (\sin^2 x)^2 \) in terms of cosine:
\[\sin^4 x = \left(\frac{1}{2}(1 - \cos(2x))\right)^2 = \frac{1}{4}(1 - \cos(2x))^2\]This technique is essential for transforming expressions to simplify them effectively, especially when asked to completely rewrite in terms of cosine. By lowering the power of both sine and cosine, identities can be combined more readily, making the expression easier to handle.

Cosine Function

The cosine function, denoted as \( \cos(x) \), is a fundamental trigonometric function. For expressions that involve powers of cosine, like \( \cos^4 x \), it's handy to use another trigonometric identity for power reduction:
\[\cos^2 x = \frac{1 + \cos(2x)}{2}\]Using this formula, a higher power such as \( \cos^4 x = (\cos^2 x)^2 \) can be expressed as:
\[\cos^4 x = \left(\frac{1 + \cos(2x)}{2}\right)^2 = \frac{1}{4}(1 + 2\cos(2x) + \cos^2(2x))\]Expanding the term \( \cos^2(2x) \) using:
\[\cos^2(2x) = \frac{1 + \cos(4x)}{2}\] allows further reduction of power until the expression is manageable. These identities help combine multiple cosine terms, ultimately reducing the overall complexity of the equation.

Sine Function

The sine function, indicated as \( \sin(x) \), is another key trigonometric function that is frequently seen in calculus and trigonometry. To transform and simplify expressions involving higher powers of sine, such as \( \sin^4 x \), we employ specific identities. As introduced, the power reduction identity provides a means to lower the sine's power:
\[\sin^2 x = \frac{1}{2}(1 - \cos(2x))\]This identity allows transformation of \( \sin^4 x \) into an expression in terms of the first power of cosine, paving the way for further simplification:
\[\sin^4 x = \frac{1}{4}(1 - 2\cos(2x) + \cos^2(2x))\]Ultimately, this method helps handle expressions where both sine and cosine functions' powers are efficiently managed, simplifying problem-solving in trigonometric computations.