Problem 53 - Contemporary Precalculus

Question

Use algebra and identities in the text to simplify the expression. Assume all denominators are nonzero. $$\left(\frac{4 \cos ^{2} t}{\sin ^{2} t}\right)\left(\frac{\sin t}{4 \cos t}\right)^{2}$$

Step-by-Step Solution

Answer

Question: Simplify the following expression, assuming all denominators are nonzero: $$\left(\frac{4 \cos ^{2} t}{\sin ^{2} t}\right)\left(\frac{\sin t}{4 \cos t}\right)^{2}$$ Solution: The simplified expression is $$\frac{4(1 - \sin^2 t)}{(4 \cos t)^{2}}$$.

Step 1: Rewrite the expression as an exponent

To start, let's rewrite the expression as an exponent: $$\left(\frac{4 \cos ^{2} t}{\sin ^{2} t}\right)\left(\frac{\sin t}{4 \cos t}\right)^{2}$$ becomes $$\frac{4 \cos ^{2} t}{\sin ^{2} t} \cdot \left(\frac{\sin ^{2} t}{(4 \cos t)^{2}}\right)$$ To do this, we applied the property \((a^m)^n = a^{mn}\) to the second part of the expression, which gives us \(\left(\frac{\sin t}{4 \cos t}\right)^{2} = \frac{\sin ^{2} t}{(4 \cos t)^{2}}\).

Step 2: Apply the Pythagorean identity

Now, let's apply the Pythagorean identity for sine and cosine: \(\sin ^{2}t + \cos ^{2}t = 1\). We can manipulate this identity to express \(\cos ^{2}t\) in terms of \(\sin ^{2}t\), like so: $$\cos ^{2}t = 1 - \sin ^{2}t$$ Substitute this expression for \(\cos ^{2}t\) in our original expression: $$\frac{4 (1 - \sin ^{2}t)}{\sin ^{2} t} \cdot \left(\frac{\sin ^{2} t}{(4 \cos t)^{2}}\right)$$

Step 3: Simplify the expression

Simplify the expression by distributing the \(4\) in the numerator of the first fraction: $$\frac{4 - 4\sin ^{2}t}{\sin ^{2} t} \cdot \left(\frac{\sin ^{2} t}{(4 \cos t)^{2}}\right)$$ Now, we can multiply the two fractions by multiplying the numerators and denominators: $$\frac{(4 - 4\sin ^{2}t)(\sin ^{2} t)}{\sin ^{2} t(4 \cos t)^{2}}$$

Step 4: Cancel out the same terms

Observe that we can cancel out the \(\sin ^{2}t\) terms from the numerator and the denominator: $$\frac{4 - 4\sin ^{2}t}{(4 \cos t)^{2}}$$

Step 5: Simplify the final expression

Finally, simplify the expression by factoring out \(4\) from the numerator: $$\frac{4(1 - \sin^2 t)}{(4 \cos t)^{2}}$$ Please note that this is the final simplified expression of our original expression: $$\left(\frac{4 \cos ^{2} t}{\sin ^{2} t}\right)\left(\frac{\sin t}{4 \cos t}\right)^{2}$$

Key Concepts

Pythagorean Identity

Understanding the Pythagorean identity is crucial when working with trigonometric expressions. This fundamental identity relates the squares of the sine and cosine functions of any angle. Specifically, it states that for any angle \( t \), the following is true:
\[ \text{Pythagorean identity: } \sin^2(t) + \cos^2(t) = 1 \]
The beauty of this identity lies in its ability to transform expressions, allowing one to replace \( \sin^2(t) \) with \( 1 - \cos^2(t) \) or \( \cos^2(t) \) with \( 1 - \sin^2(t) \), thereby simplifying complex trigonometric expressions. This substitution is a game-changer for simplification because it often clears the way for further algebraic manipulation, leading to a more streamlined expression.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. They are indispensable tools for simplifying and solving trigonometric equations. Apart from the Pythagorean identity, there are several other key identities that students should familiarize themselves with:

• Reciprocal identities like \( \frac{1}{\sin(t)} = \csc(t) \) and \( \frac{1}{\cos(t)} = \sec(t) \).
• Quotient identities such as \( \frac{\sin(t)}{\cos(t)} = \tan(t) \) and \( \frac{\cos(t)}{\sin(t)} = \cot(t) \).
• Co-function identities which relate functions to complementary angles, for instance, \( \sin(\frac{\pi}{2} - t) = \cos(t) \).

By strategically applying these identities, you can simplify trigonometric expressions and solve equations more efficiently. They serve as the building blocks for deeper understanding in trigonometry, opening up pathways to solving more complex problems.

Algebraic Manipulation

Algebraic manipulation involves the use of algebraic rules to rearrange and simplify equations or expressions. This can include operations such as factoring, expanding, or canceling terms. In the context of trigonometry, algebraic manipulation often works hand-in-hand with trigonometric identities to simplify expressions. A common process involves factoring out common terms, canceling out like terms in the numerator and denominator, or simplifying complex fractions.
It's important to recognize opportunities for simplification. For instance, when similar terms appear in both the numerator and the denominator, they may be divided out. Also, looking for common factors, as in the step where \( 4 \) is factored out of the numerator, can lead to a more streamlined result. Mastery of algebraic manipulation not only aids in solving trigonometric problems but also enhances overall mathematical problem-solving skills.