Problem 41 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-50: Verify the identity. $$ \left(1-\tan ^{2} \phi\right)^{2}=\sec ^{4} \phi-4 \tan ^{2} \phi $$

Step-by-Step Solution

Answer

The identity \((1-\tan^2 \phi)^2 = \sec^4 \phi - 4\tan^2 \phi\) is verified.

Step 1: Recognize trigonometric identities

Recall that the tangent and secant functions are related through the identity \( \sec^2 \phi = 1 + \tan^2 \phi \). We will use this identity to simplify the given equation.

Step 2: Expand the left side

Expand \( (1 - \tan^2 \phi)^2 \) using the formula for the square of a binomial: \( (a - b)^2 = a^2 - 2ab + b^2 \). This results in: \[ (1 - \tan^2 \phi)^2 = 1 - 2\tan^2 \phi + \tan^4 \phi. \]

Step 3: Simplify the right side

Substitute \( \sec^2 \phi \) in the right side of the equation using \( \sec^2 \phi = 1 + \tan^2 \phi \) to express \( \sec^4 \phi \) as:\[ \sec^4 \phi = (1 + \tan^2 \phi)^2. \]

Step 4: Expand \( \sec^4 \phi \)

Now expand \( (1 + \tan^2 \phi)^2 \): \[ (1 + \tan^2 \phi)^2 = 1 + 2\tan^2 \phi + \tan^4 \phi. \]

Step 5: Substitute and compare

Substitute the expansion from Step 4 into the simplified right side equation:\[ \sec^4 \phi - 4\tan^2 \phi = 1 + 2\tan^2 \phi + \tan^4 \phi - 4\tan^2 \phi. \]Simplify this to:\[ 1 - 2\tan^2 \phi + \tan^4 \phi. \]

Step 6: Verify the identity

Compare the simplified expressions from the left and right sides:* Left side: \( 1 - 2\tan^2 \phi + \tan^4 \phi \)* Right side: \( 1 - 2\tan^2 \phi + \tan^4 \phi \)Since both expressions are identical, the identity is verified.

Key Concepts

Tangent Function

The tangent function (\( \tan \phi \)) is one of the basic trigonometric functions. It is often used in various mathematical and engineering fields. Understanding the tangent function is essential as it helps in solving different problems involving angles and ratios.

• Definition: The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.
• Expression: Mathematically, \( \tan \phi = \frac{\text{Opposite}}{\text{Adjacent}} \).

The tangent function is periodic, meaning it repeats values over regular intervals. This makes it useful in modeling cyclical phenomena. An important property of the tangent is its relation to the secant function, which states: \( \sec^2 \phi = 1 + \tan^2 \phi \).This relationship is pivotal in verifying identities, as we have seen in the provided exercise.

Secant Function

The secant function is another fundamental trigonometric function. It has a critical role especially when it comes to verifying identities involving tangent.

• Definition: The secant of an angle is the reciprocal of the cosine function, which is \( \sec \phi = \frac{1}{\cos \phi} \).
• Expression: It is closely related to the tangent function through the identity \( \sec^2 \phi = 1 + \tan^2 \phi \).

The secant function is used in situations where one needs to avoid dividing by zero in certain trigonometric equations. In addition, \( \sec^4 \phi \) can be expanded to help in breaking down complex equations, as appreciated when manipulating expressions as seen in the solution steps.

Algebraic Manipulation

Algebraic manipulation is an essential skill in mathematics, especially when dealing with trigonometric identities. These manipulations allow us to transform and simplify expressions, as we did in solving the given exercise.

• Expansion: Using identities like \((a-b)^2 = a^2 - 2ab + b^2\) helps in expanding expressions.
• Substitution: Replacing parts of an equation using known identities such as \( \sec^2 \phi = 1 + \tan^2 \phi \) is a powerful tool.

These techniques are crucial for verifying trigonometric identities and solving trigonometric equations. Practicing algebraic manipulations can significantly improve problem-solving speed and accuracy.

Trigonometric Equations

Trigonometric equations involve values of trigonometric functions of unknown angles. Solving them requires an understanding of the basic trigonometric functions and identities.

• Solving Process: Often involves substitution and algebraic manipulation to transform the equations into a more solvable form.
• Verification: After solving, it's crucial to verify that solutions satisfy the original equation.

Understanding the intangibles of trigonometric equations helps in applying these skills to real-world scenarios, such as physics and engineering problems. Verifying identities, like the one in the exercise, requires comparing both sides of the equation after appropriate simplifications, ensuring they are equivalent, thus proving the identity.