Problem 25 - Precalculus: Functions and Graphs

Question

Verify the Identity. $$(\sec t+\tan t)^{2}=\frac{1+\sin t}{1-\sin t}$$

Step-by-Step Solution

Answer

The identity is verified: both sides simplify to \(\frac{(1 + \sin t)^2}{\cos^2 t}\).

Step 1: Expand the Left Side

Begin by expanding the left side of the identity \[(\sec t + \tan t)^{2} = (\sec t)^{2} + 2(\sec t)(\tan t) + (\tan t)^{2} \]Remember, \((a+b)^{2} = a^{2} + 2ab + b^{2}\).

Step 2: Use Trigonometric Identities

Substitute \(\sec t = \frac{1}{\cos t}\) and \(\tan t = \frac{\sin t}{\cos t}\) into the expanded expression.This becomes:\[(\frac{1}{\cos t})^{2} + 2 \cdot \frac{1}{\cos t} \cdot \frac{\sin t}{\cos t} + (\frac{\sin t}{\cos t})^{2}\]which simplifies to:\[\frac{1}{\cos^2 t} + \frac{2 \sin t}{\cos^2 t} + \frac{\sin^2 t}{\cos^2 t}\]

Step 3: Simplify the Left Side Expression

Combine the terms over the common denominator \(\cos^2 t\):\[\frac{1 + 2 \sin t + \sin^2 t}{\cos^2 t}\]Notice that \(1 + \sin^2 t = (1 + \sin t)^2 - 2\sin t\). Substitute this to simplify further:\[\frac{(1 + \sin t)^2}{\cos^2 t}\]

Step 4: Simplify the Right Side

The right side of the equation is:\[\frac{1 + \sin t}{1 - \sin t}\]Multiply numerator and denominator by \((1 + \sin t)\):\[\frac{(1 + \sin t)(1 + \sin t)}{(1 - \sin t)(1 + \sin t)}\]This simplifies to:\[\frac{(1 + \sin t)^2}{1 - \sin^2 t}\]Using the identity \(1 - \sin^2 t = \cos^2 t\), the expression becomes:\[\frac{(1 + \sin t)^2}{\cos^2 t}\]

Step 5: Verify the Identity

Both the simplified expressions from the left and right sides of the original equation equal:\[\frac{(1 + \sin t)^2}{\cos^2 t}\]Thus, the identity \((\sec t + \tan t)^{2} = \frac{1 + \sin t}{1 - \sin t}\) is verified.

Key Concepts

Verify Identities

In trigonometry, verifying identities involves proving that two different trigonometric expressions are equal for all values of the variables involved. In this exercise, we were tasked with verifying the identity \[(\sec t + \tan t)^{2} = \frac{1 + \sin t}{1 - \sin t}\]This means we need to confirm that both sides of the equation are equivalent through algebraic manipulation and trigonometric identities.

To verify such identities, follow these steps:

• Start by choosing one side of the equation to manipulate. Typically, this involves expanding or simplifying using known trigonometric identities.
• Employ various identities, like the Pythagorean identities or reciprocal identities, to transform complex expressions into simpler forms.
• Continue simplifying until you can prove both sides are algebraically equivalent.

Verifying identities is fundamental in trigonometry, as it aids in understanding the relationships between different trigonometric functions.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, secant, and tangent, are the cornerstone of trigonometry. They relate the angles and sides of a triangle, especially in the context of a right-angled triangle.

In this exercise:

• Secant function (\(\sec\)): It is the reciprocal of the cosine function. So, \(\sec t = \frac{1}{\cos t}\).
• Tangent function (\(\tan\)): Defined as the ratio of sine to cosine, hence \(\tan t = \frac{\sin t}{\cos t}\).

When working with problems involving these functions, it is crucial to be familiar with how they can be transformed and applied through identities. For instance, using these definitions allows us to express complex trigonometric polynomials in terms of simpler expressions, as shown in the step-by-step solution.

Simplifying Expressions

Simplifying trigonometric expressions is about reducing them to their simplest and most manageable form. This often involves applying fundamental identities and algebraic techniques that allow for the combination and reduction of terms into a cleaner representation.

For example, with the expression \[\frac{1 + 2 \sin t + \sin^2 t}{\cos^2 t}\] from the solution, we simplified it by recognizing patterns that resemble known identities. The expression \[1 + \sin^2 t\] was rewritten using the fact that \[(1 + \sin t)^2 = 1 + 2\sin t + \sin^2 t\], transforming it into a perfect square format.

Takeaways when simplifying:

• Always look out for factorable expressions, squares, and sums that align with identities.
• Remember, the goal is to reach an equivalent but simpler form that readily reveals the identity.
• Avoid over-complicating steps; seek the most direct approach in simplifying.