Problem 54 - Precalculus Mathematics for Calculus

Question

Verify the identity. $$\sin ^{2} \alpha+\cos ^{2} \alpha+\tan ^{2} \alpha=\sec ^{2} \alpha$$

Step-by-Step Solution

Answer

The identity is verified: \( \sin^2 \alpha + \cos^2 \alpha + \tan^2 \alpha = \sec^2 \alpha \).

Step 1: Use the Pythagorean Identity for Sine and Cosine

Recall that the Pythagorean identity is \( \sin^2 \alpha + \cos^2 \alpha = 1 \). This identity will be useful as part of our simplification process.

Step 2: Express Tan and Sec in Terms of Sin and Cos

The tangent of \( \alpha \) is defined as \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \) and the secant is \( \sec \alpha = \frac{1}{\cos \alpha} \). These will help us rewrite the given identity.

Step 3: Substitute Tan Identity into the Left Side

Substitute \( \tan^2 \alpha \) in the equation with \( \left(\frac{\sin \alpha}{\cos \alpha}\right)^2 \) to rewrite the expression as \( \sin^2 \alpha + \cos^2 \alpha + \frac{\sin^2 \alpha}{\cos^2 \alpha} \).

Step 4: Simplify the Expression

Using the identity from Step 1, substitute 1 for \( \sin^2 \alpha + \cos^2 \alpha \) in the expression to get \( 1 + \frac{\sin^2 \alpha}{\cos^2 \alpha} \).

Step 5: Combine the Fractions

The expression \( 1 + \frac{\sin^2 \alpha}{\cos^2 \alpha} \) can be written as \( \frac{\cos^2 \alpha}{\cos^2 \alpha} + \frac{\sin^2 \alpha}{\cos^2 \alpha} \). Combine the fractions to get \( \frac{\cos^2 \alpha + \sin^2 \alpha}{\cos^2 \alpha} \).

Step 6: Simplify the Combined Fraction

Since \( \cos^2 \alpha + \sin^2 \alpha = 1 \), substitute 1 for the numerator. The fraction now is \( \frac{1}{\cos^2 \alpha} \).

Step 7: Recognize the Result as Sec Identity

Notice that \( \frac{1}{\cos^2 \alpha} \) is \( \sec^2 \alpha \). This proves that the left side of the identity simplifies to the right side, \( \sec^2 \alpha \).

Key Concepts

Pythagorean identity

The Pythagorean identity is one of the most well-known and widely used identities in trigonometry. It states that for any angle \( \alpha \), the square of the sine of \( \alpha \) added to the square of the cosine of \( \alpha \) equals one. In mathematical terms, this is written as: \[\sin^2 \alpha + \cos^2 \alpha = 1.\] This identity is incredibly helpful when solving trigonometric equations because it allows you to relate sine and cosine in a simple equation.

• The identity helps convert expressions involving squares of sine and cosine into simpler forms.
• It forms the basis for many other trigonometric identities and equations.

In this exercise, the Pythagorean identity is used to eliminate \( \sin^2 \alpha + \cos^2 \alpha \) from the equation, simplifying it to 1. This simplification plays a crucial role in eventually proving the original identity provided in the exercise.

tangent function

The tangent function is another essential part of trigonometry, represented as \( \tan \alpha \). It is defined as the ratio of the sine of an angle to the cosine of that angle: \[\tan \alpha = \frac{\sin \alpha}{\cos \alpha}.\] This definition is useful for various purposes, such as transforming identities or solving equations involving tangents. It provides a way to express one trigonometric function (tangent) entirely in terms of two others (sine and cosine).

• The tangent function makes it possible to express complex functions involving sine and cosine in simpler forms.
• For this particular exercise, substituting \( \tan^2 \alpha \) as \( \left(\frac{\sin \alpha}{\cos \alpha}\right)^2 \) helps move from the original expression to a form that can be easily simplified using algebraic techniques.

By substituting and manipulating these expressions, you can merge trigonometric functions to find simplified identities.

secant function

The secant function is the reciprocal of the cosine function. It is written as \( \sec \alpha \) and defined by the equation: \[\sec \alpha = \frac{1}{\cos \alpha}.\] The secant function is vital in trigonometry for creating and proving various identities, including the one you see in this exercise. By understanding how \( \sec \alpha \) relates to \( \cos \alpha \), you can unlock new ways to reframe trigonometric expressions.

• Using secant can help transition a problem into simpler forms by changing division-based expressions into multiplication.
• In the exercise, the goal was to establish that the original expression simplifies to \( \sec^2 \alpha \), thus proving the given identity.

Recognizing \( \sec^2 \alpha \) as \( \frac{1}{\cos^2 \alpha} \) completes the transformation, showing how trigonometric identities are often linked through simple mathematical relationships.