Problem 47 - Technical Mathematics with Calculus

Question

Prove each identity. (All identities in this chapter can be proven. ) $$\frac{\sin \theta+1}{1-\sin \theta}=(\tan \theta+\sec \theta)^{2}$$

Step-by-Step Solution

Answer

By converting \(\tan \theta \) and \(\sec \theta\) into \(\sin \theta\) and \(\cos \theta\), expanding, applying the Pythagorean identity, and simplifying, the identity \(\frac{\sin \theta+1}{1-\sin \theta}=(\tan \theta+\sec \theta)^{2}\) is proven.

Step 1: Convert the Right Side to Sine and Cosine

Begin by expressing \(\tan \theta\) as \(\frac{\sin \theta}{\cos \theta}\) and \(\sec \theta\) as \(\frac{1}{\cos \theta}\). This gives us \((\frac{\sin \theta}{\cos \theta} + \frac{1}{\cos \theta})^2\) on the right side of the equation.

Step 2: Combine Terms on the Right Side

Combine the terms inside the parentheses by putting them over the same denominator which is \(\cos \theta\), then square the resulting term. This results in \(\left(\frac{\sin \theta + 1}{\cos \theta}\right)^2\).

Step 3: Square the Numerator and Denominator

Square both the numerator and the denominator to get \(\frac{(\sin \theta + 1)^2}{\cos^2 \theta}\).

Step 4: Expand the Squared Numerator

Expand the binomial in the numerator to get \(\frac{\sin^2 \theta + 2\sin \theta + 1}{\cos^2 \theta}\).

Step 5: Utilize the Pythagorean Identity

Apply the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to the numerator. This transforms it into \(\frac{1 - \cos^2 \theta + 2\sin \theta + 1}{\cos^2 \theta}\).

Step 6: Simplify the Numerator

Simplify the numerator by replacing \(1 - \cos^2 \theta\) with \(\sin^2 \theta\) to get \(\frac{2\sin \theta + 2}{\cos^2 \theta}\).

Step 7: Factor Out the Common Term

Factor out 2 from the numerator to get \(\frac{2(\sin \theta + 1)}{\cos^2 \theta}\).

Step 8: Express \(\cos^2 \theta\) as \((1 - \sin \theta)(1 + \sin \theta)\)

Through the difference of squares, \(\cos^2 \theta\) is equivalent to \((1 - \sin \theta)(1 + \sin \theta)\). Write the expression as \(\frac{2(\sin \theta + 1)}{(1 - \sin \theta)(1 + \sin \theta)}\).

Step 9: Cancel the Common Terms

The term \((\sin \theta + 1)\) in the numerator and the term \((1 + \sin \theta)\) in the denominator are exactly the same and can be cancelled out, leaving \(\frac{2}{1 - \sin \theta}\).

Step 10: Simplify the Result

Finally, cancel the factor of 2 with the denominator to obtain the left side of the original identity, \(\frac{\sin \theta + 1}{1 - \sin \theta}\), thus proving the identity.

Key Concepts

Trigonometric Identity

Trigonometric identities are mathematical expressions that establish the relationship between trigonometric functions. They play a crucial role in simplifying complex trigonometric equations and proving that two expressions are equivalent.

For example, when working with the identity \[\frac{\sin \theta+1}{1-\sin \theta}=(\tan \theta+\sec \theta)^{2}\], it is imperative to manipulate both sides of the equation strategically to demonstrate that they are indeed the same. In our case, converting tangent and secant to sine and cosine on the right side is the first step, as these fundamental trigonometric functions often form the basis for more complex identities.

Pythagorean Identity

The Pythagorean identity, \[\sin^2 \theta + \cos^2 \theta = 1\], is one of the most important and widely recognized trigonometric identities. It stems from the Pythagorean theorem related to right triangles. When proving trigonometric identities, such as in our textbook example, this identity allows us to interchange \(\sin^2 \theta\) with \(1 - \cos^2 \theta\), and vice versa, facilitating the simplification process.

As seen in the solution, by applying the Pythagorean identity, we can transform the numerator and move one step closer to proving the initial identity.

Sine and Cosine

The sine and cosine functions are the fundamental trigonometric functions, defined for a right-angle triangle as the ratios of specific sides. In the unit circle context, sine represents the y-coordinate and cosine represents the x-coordinate of a point on the circle.

Understanding the definitions and properties of these functions is key when proving trigonometric identities. As shown in our example, manipulating the right side of the equation to be in terms of sine and cosine makes the proving process more straightforward, as these forms often lead to cancellations or the application of the Pythagorean identity.

Binomial Expansion

Binomial expansion involves expanding expressions that are raised to a power, such as \(a + b)^n\). For\[ (\sin \theta + 1)^2 \], we use binomial expansion to get \[\sin^2 \theta + 2\sin \theta + 1\].

This step turns the squared term into a form where the Pythagorean identity can be applied, thus revealing deeper connections within the expression. In our exercise, expanding the squared numerator is a vital step to reaching the simple form that matches the left side of the original identity.