Problem 52 - Algebra and Trigonometry

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 as true.

Step 1: Recall the Pythagorean Identity

The Pythagorean identity states that \( \sin^2 \alpha + \cos^2 \alpha = 1 \). This is a fundamental trigonometric identity and should be remembered.

Step 2: Express \( \tan^2 \alpha \) in terms of \( \sec^2 \alpha \)

The identity for tangent and secant is \( \tan^2 \alpha = \sec^2 \alpha - 1 \). This is derived from \( \sec^2 \alpha = 1 + \tan^2 \alpha \).

Step 3: Substitute \( \tan^2 \alpha \) and Simplify

Substitute the expression for \( \tan^2 \alpha \) from Step 2 into the original equation: \( \sin^2 \alpha + \cos^2 \alpha + (\sec^2 \alpha - 1) = \sec^2 \alpha \). Simplify this to \( 1 + \sec^2 \alpha - 1 = \sec^2 \alpha \), which reduces to the identity \( \sec^2 \alpha = \sec^2 \alpha \).

Step 4: Conclude the Verification

Since the equation simplifies to \( \sec^2 \alpha = \sec^2 \alpha \), which is trivially true, the original expression is indeed an identity.

Key Concepts

Pythagorean identity

The Pythagorean identity is a cornerstone in trigonometry, deeply rooted in the Pythagorean theorem. It is expressed as \( \sin^2 \alpha + \cos^2 \alpha = 1 \). This equation reflects the fundamental relationship between the sine and cosine functions in a right triangle, where the square of the hypotenuse equals the sum of the squares of the other two sides. In the context of the unit circle, this identity states that for any angle \( \alpha \), the sum of the squares of the x-coordinate (cosine) and the y-coordinate (sine) of the point on the circle is always 1.
This identity is crucial as it provides a simple yet powerful tool for solving various trigonometric problems.
It helps in simplifying complex expressions and verifying other identities. When working with trigonometric expressions, always remember that applying this identity can transform and simplify equations significantly.

tangent and secant relationship

The tangent and secant relationship is another essential trigonometric identity that connects these two functions. It is given by \( \sec^2 \alpha = 1 + \tan^2 \alpha \). This identity emerges from the division of the Pythagorean identity by \( \cos^2 \alpha \). When you divide \( \sin^2 \alpha + \cos^2 \alpha = 1 \) by \( \cos^2 \alpha \), you obtain \( \tan^2 \alpha + 1 = \sec^2 \alpha \).
This is because \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \) and \( \sec \alpha = \frac{1}{\cos \alpha} \).
Understanding this relationship is vital, particularly when simplifying expressions involving tangent and secant. It helps transform complex trigonometric expressions into simpler forms, easier to handle. This identity also allows for the rearrangement of equations making the process of verification or simplification more straightforward.

trigonometric simplification

Trigonometric simplification involves the use of identities and algebraic manipulation to reduce trigonometric expressions to their simplest form. In the exercise provided, the simplification leverages the Pythagorean identity and the tangent-secant relationship. By substituting \( \tan^2 \alpha \) with \( \sec^2 \alpha - 1 \), the expression becomes easier to manage.
This approach is crucial in verifying identities, solving equations, and facilitating deeper understanding of trigonometric concepts.
Here are some general steps for successful trigonometric simplification:

• Identify applicable identities (e.g., Pythagorean, angle sum, or tangent-secant relationships).
• Look for opportunities to factor or combine like terms.
• Substitute known values from identities to reduce complexity.
• Continue rearranging until you reach the simplest form.

Trigonometric simplification is an invaluable skill that enhances problem-solving efficiency and accuracy.