Problem 16 - Algebra and Trigonometry

Question

Verify each identity. \(\cos ^{2} \theta\left(1+\tan ^{2} \theta\right)=1\)

Step-by-Step Solution

Answer

After simplifying and applying the trigonometric identities appropriately, we can show that \(\cos^{2}\theta\left(1+\tan^{2}\theta\right)\) simplifies to 1, which verifies the given identity.

Step 1: Understanding and Rearranging the Identity

Firstly, let's understand the given trigonometric identity: \(\cos^{2}\theta\left(1+\tan^{2}\theta\right)=1\). The main task is to simplify the left side and show it equals to the right side, which is 1. Start by using the identity \(\tan^{2}\theta= \frac{\sin^{2}\theta}{\cos^{2}\theta}\). So, the left side will be rewritten as: \(\cos^{2}\theta(1+ \frac{\sin^{2}\theta}{\cos^{2}\theta})\)

Step 2: Simplifying Further

Continue simplifying the obtained expression: \(\cos^{2}\theta(\frac{\cos^{2}\theta + \sin^{2}\theta}{\cos^{2}\theta})\). We can simplify further by dividing each term inside the parenthesis by \(\cos^2\theta\), providing: \(1+ \frac{\sin^{2}\theta}{\cos^{2}\theta}=1+\tan^{2}\theta\)

Step 3: Return to Original Identity

The obtained result from step 2 can be related back to the original identity. Let's rewrite the obtained result: \(1+ \tan^{2}\theta\), as \(\cos^{2}\theta \left(1+ \tan^{2}\theta\right)\). This is exactly the left side of the original identity. Therefore, we have shown that the left side of the original identity equals to the right side, hence verifying the identity.

Key Concepts

Understanding Cosine in Trigonometric Identities

Cosine, often abbreviated as cos, is an integral part of trigonometric identities. It is one of the primary trigonometric functions and represents the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle. When encountering a trigonometric identity involving cosine, like \[ \cos^2 \theta (1 + \tan^2 \theta) = 1 \]it is crucial to understand how cosine interacts with other trigonometric functions.

• Cosine squared, \( \cos^2 \theta \), implies the square of the cosine function.
• This can often appear in identities to facilitate simplification.

In the context of identities, cosine is frequently paired with sine, through the fundamental identity \( \cos^2 \theta + \sin^2 \theta = 1 \). Recognizing and manipulating this relationship is key to simplifying and verifying identities. By understanding how cosine operates and its relationship with sine, you can solve many trigonometric problems efficiently.
Cosine plays a pivotal role in converting complex looking identities into simpler forms, leading us toward their verification.

Exploring Tangent in Trigonometric Relationships

Tangent, denoted as \( \tan \), is another fundamental trigonometric function. It represents the ratio of the length of the opposite side to the adjacent side in a right triangle. Tangent has a special place in many trigonometric identities. In the identity given:\[ \cos^2 \theta \left(1 + \tan^2 \theta\right) = 1 \]We use the relationship \( \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \) to simplify expressions involving tangent.

• This relationship transforms equations by replacing tangent with sine and cosine, which are often easier to work with.
• Recognizing and applying this identity helps in breaking down and verifying complex trigonometric expressions.

Understanding how tangent relates to sine and cosine is essential for verifying identities, as seen where \( 1 + \tan^2 \theta \) is simplified using the Pythagorean trigonometric identity \( \cos^2 \theta + \sin^2 \theta = 1 \).
By mastering the role and behavior of tangent, you can seamlessly transition through trigonometric equations and confirm their validity.

Delving into Sine and its Trigonometric Applications

Sine, which we write as \( \sin \), completes the trio of primary trigonometric functions. It measures the ratio of the opposite side to the hypotenuse in a right triangle. When involved in identities, sine is often seen squared, as in \( \sin^2 \theta \), or interacting with cosine as part of the Pythagorean identity. In the expression:\[ \cos^2 \theta\left(1 + \frac{\sin^2 \theta}{\cos^2 \theta}\right) \]sine appears in the form of \( \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \), showcasing the interplay between these trigonometric identities.

• Sine, alongside cosine, helps conserve identity forms, simplifying expressions dramatically.
• Sine's primary role is often in conjunction with other functions, like cosine and tangent, to establish known identities and verify new ones.

When working with sine, always consider its relationship with cosine, as in the identity \( \cos^2 \theta + \sin^2 \theta = 1 \), to guide simplification and verification processes. Understanding these relationships solidifies the foundation for working with any trigonometric expression.