Problem 46 - Algebra and Trigonometry Real Mathematics, Real People

Question

Verify the identity algebraically. Use the table feature of a graphing utility to check your result numerically. $$\sec ^{2} y-\cot ^{2}\left(\frac{\pi}{2}-y\right)=1$$

Step-by-Step Solution

Answer

The given identity is correct and can be verified using trigonometric identities. The algebraic proof leads us to \(\tan^2(y) = 1\) while graphical proof will depict two overlapping patterns, further concreting this fact.

Step 1: Utilize Pythagorean Identity

First, recall the following Pythagorean identity for secant: \(\sec^2(y) = 1 + \tan^2(y)\). Substitute this into the given identity, which gives us \(1 + \tan^2(y) - \cot^2(\frac{\pi}{2} - y) = 1\). The '1' on both sides will cancel out.

Step 2: Apply Reciprocal Identity

Consider the reciprocal identity \(\cot (x) = \frac{1}{\tan (x)}\) and apply it to the existing equation. We then have \(\tan^2(y) - \left(\frac{1}{\tan(\frac{\pi}{2} - y)}\right)^2 = 0\).

Step 3: Apply Co-Function Identity

Utilize co-function identity \(\tan (\frac{\pi}{2} - y) = \cot(y)\) thereby altering the equation to \(\tan^2(y) - {\cot^2(y)} = 0\). This step altogether readjusts the equation to \(\tan^2(y) - \frac{1}{\tan^2(y)} = 0\).

Step 4: Simplify the equation

The equation \(\tan^2(y) - \frac{1}{\tan^2(y)} = 0\) can be rewritten as \(\tan^4(y) - 1 = 0\) which is a difference between squares. This can also be written as \((\tan^2(y) - 1)(\tan^2(y) + 1) = 0\). This implies that either \(\tan^2(y) = 1\) or \(\tan^2(y) = -1\). However, since the square of a real number cannot be negative, we discard \(\tan^2(y) = -1\). Hence, \(\tan^2(y) = 1\) is the only valid equation here.

Step 5: Finding Numerical Verification

To confirm this algebraically derived result, a graphing utility can be used to plot both sides of the initial identity as functions of \(y\). If both graphs follow the same trend, it proves the identity is correct numerically as well.

Key Concepts

Pythagorean Identity

Understanding the Pythagorean identity is critical when studying trigonometry. It characterizes the fundamental relationship between the sine and cosine of an angle. The Pythagorean identity states that for any angle \theta, the following holds true: \( \. \sin^2(\theta) + \cos^2(\theta) = 1 \).

For the secant function, which is the reciprocal of the cosine, the Pythagorean identity transforms into \( \. \sec^2(y) = 1 + \tan^2(y) \), where \tan(y)\ is the tangent of \y\. This relationship was used in Step 1 of the solution to substitute \sec^2(y)\ with \(1 + \tan^2(y)\) in our verification process. This substitution is the foundation for verifying the original trigonometric identity.

A way to visualize this is to consider a right-angled triangle. The lengths of the sides in relation to the angles are directly described by the Pythagorean theorem, which in turn informs these trigonometric identities. Whenever you come across \sec^2\ or \tan^2\ in an identity, remember that they are deeply connected through the Pythagorean theorem.

Reciprocal Trigonometric Identity

Reciprocal identities are also a cornerstone of trigonometry, demonstrating how trigonometric functions can be expressed in terms of each other. For instance, the reciprocal identity for the cotangent states that \( \cot(x) = \frac{1}{\tan(x)} \). Essentially, it implies that the cotangent of an angle is the multiplicative inverse of the tangent of that angle.

In Step 2 of the solution, this identity was vital to reshape the equation from \( \cot^2( \frac{\pi}{2} - y) \) to a tangent form. This allowed for further simplification. Remember that reciprocal identities will often allow you to express a trigonometric function in a more convenient form for the problem at hand. It's always a good practice to look for opportunities to apply these identities early in the simplification process.

Co-Function Trigonometric Identity

Lastly, co-function identities are a set of equations that relate certain pairs of trigonometric functions evaluated at complementary angles. For example, the identity \( \tan \left( \frac{\pi}{2} - y \right) = \cot(y) \) demonstrates the relationship between the tangent and cotangent functions at complementary angles.

During Step 3 of the problem-solving process, the co-function identity was used to convert \( \tan^2(y) - \(\frac{1}{\tan(\frac{\pi}{2} - y)}\)^2 = 0 \) into \( \tan^2(y) - \. \cot^2(y) = 0 \.\), thanks to the complementary angle relationship. The ability to recognize when to use a co-function identity can significantly simplify the process of verifying trigonometric identities, as it can transform an equation into a more familiar or workable form.

Overall, familiarizing yourself with these three core identities — Pythagorean, reciprocal, and co-function — equips you with powerful tools for verifying and simplifying a wide range of trigonometric expressions, thus helping you to tackle complex trigonometric challenges more effectively.