Problem 58 - Algebra 2

Question

Verify each identity. $$ \sin ^{2} \theta \tan ^{2} \theta+\cos ^{2} \theta \tan ^{2} \theta=\sec ^{2} \theta-1 $$

Step-by-Step Solution

Answer

The given trigonometric identity is valid. After manipulation, both sides of the equation simplifies to \(\frac{\sin^{2}\theta}{\cos^{2}\theta}\), thus verifying the identity.

Step 1: Convert the Trigonometric Functions

To begin solving this identity, it's useful to convert all trigonometric terms to terms of sine and cosine, since these are the basic building blocks of trigonometry. Starting from the left side, we can rewrite \(\tan^{2}\theta\) as \(\frac{\sin^{2}\theta}{\cos^{2}\theta}\), so the left side becomes: \(\sin^{2}\theta \frac{\sin^{2}\theta}{\cos^{2}\theta} + \cos^{2}\theta \frac{\sin^{2}\theta}{\cos^{2}\theta}\)

Step 2: Simplify the Left Side

By cancelling out the denominators, the above expression simplifies to: \(\sin^{4}\theta + \sin^{2}\theta\)

Step 3: Rewrite the Right Side

Now, we attend the right side of the identity using the relationship between secant and cosine functions: \(\sec^{2}\theta - 1 = \frac{1}{\cos^{2}\theta} - 1\)

Step 4: Simplify the Right Side

Simplify by finding a common denominator: \( \frac{1}{\cos^{2}\theta} - 1 = \frac{1 - \cos^{2}\theta}{\cos^{2}\theta}\). And by using the Pythagorean identity \(\sin^{2}\theta + \cos^{2}\theta = 1\), the latter equation simplifies to: \(\frac{\sin^{2}\theta}{\cos^{2}\theta}\)

Step 5: Verification

Now, both the left side and the right side of the equation are equal, which verifies the given trigonometric identity.

Key Concepts

Sine and Cosine Functions

The sine and cosine functions are fundamental in trigonometry and form the basis for many other trigonometric expressions and identities. These functions stem from the Greek term of 'circle measurement' and are used primarily to relate the angles and lengths of right-angled triangles.

• The sine function, denoted as \( \sin \theta \), represents the ratio of the length of the side opposite the angle \( \theta \) to the hypotenuse in a right triangle.
• The cosine function, denoted as \( \cos \theta \), is the ratio of the length of the adjacent side to the hypotenuse.

These two functions can also be visualized on the unit circle, a concept used to extend their definitions beyond 90 degrees. On the unit circle, the sine of an angle is the \( y \)-coordinate of the corresponding point, and the cosine is the \( x \)-coordinate. This makes sine and cosine integral for defining other trigonometric functions and solving trigonometric identities, such as converting expressions to simpler forms as seen in the exercise above.

Tangent Function

The tangent function, another essential trigonometric function, is a combination of sine and cosine functions. It is represented as \( \tan \theta \) and can be defined as the ratio of sine to cosine:\[\tan \theta = \frac{\sin \theta}{\cos \theta}\]This makes tangent a powerful tool for calculations, especially when working with angles and triangles.

• In terms of the right triangle, it represents the ratio of the length of the opposite side to the adjacent side.
• Tangent is undefined when \( \cos \theta = 0 \), as division by zero is not possible, typically occurring at odd multiples of \( \frac{\pi}{2} \).

Understanding the tangent function's relationship to sine and cosine allows us to transform complex trigonometric expressions, which aids in solving problems like the provided exercise, where tangent squared \( \tan^2 \theta \) was broken into its sine and cosine components to verify the identity.

Secant Function

The secant function, denoted as \( \sec \theta \), is less frequently mentioned when compared to sine, cosine, and tangent but is equally significant. It is the reciprocal of the cosine function:\[\sec \theta = \frac{1}{\cos \theta}\]Secant is particularly useful when dealing with identities and solving trigonometric equations. Because it is the reciprocal of cosine, secant will be undefined wherever cosine equals zero.

• It helps simplify trigonometric expressions, as seen in the step-by-step solution where \( \sec^2 \theta - 1 \) was used on the right side of the equation.
• Secant is infrequent in basic trigonometry, but mastering it eventually proves beneficial for more advanced problems, especially when working with Pythagorean Identities and transformations.

In our exercise, leveraging the secant function allowed us to transition expressions and simplify the equation efficiently.

Pythagorean Identity

The Pythagorean Identity is one of the most well-known and vital trigonometric identities. It can be derived from the Pythagorean Theorem and relates the squares of sine and cosine functions:\[\sin^2 \theta + \cos^2 \theta = 1\]This identity is a cornerstone in trigonometry, permitting the transformation and simplification of expressions.

• This equation allows us to substitute expressions involving \( \sin^2 \theta \) or \( \cos^2 \theta \) with each other, which is particularly useful in proofs and verification of identities.
• It can also be rearranged to find expressions for \( \cos^2 \theta \) or \( \sin^2 \theta \) in terms of the other, greatly enhancing our toolset for solving equations.

In the exercise outlined, the Pythagorean Identity assists in converting and simplifying both sides of the equation, specifically reshaping \( 1 - \cos^2 \theta \) into \( \sin^2 \theta \). This flexibility is what makes the Pythagorean Identity irreplaceable in solving trigonometric problems.