Problem 84 - Precalculus Mathematics for Calculus

Question

Prove the identity. $$\tan ^{2}\left(\frac{x}{2}+\frac{\pi}{4}\right)=\frac{1+\sin x}{1-\sin x}$$

Step-by-Step Solution

Answer

The identity is proven: \( \tan^2\left(\frac{x}{2}+\frac{\pi}{4}\right)=\frac{1+\sin x}{1-\sin x} \).

Step 1: Express the tangent function in sine and cosine

The tangent function in terms of sine and cosine is given by \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). Apply it for \( \theta = \frac{x}{2} + \frac{\pi}{4} \).

Step 2: Formula Substitution

Using the identity for \( \tan(\theta) \):\[\tan^2\left(\frac{x}{2} + \frac{\pi}{4}\right) = \left(\frac{\sin\left(\frac{x}{2} + \frac{\pi}{4}\right)}{\cos\left(\frac{x}{2} + \frac{\pi}{4}\right)}\right)^2\]Express \(\sin\) and \(\cos\) using angle sum identities.

Step 3: Apply angle sum identities

The angle sum identities are:\[\sin(a + b) = \sin a \cos b + \cos a \sin b, \\cos(a + b) = \cos a \cos b - \sin a \sin b\]Apply these:\[\sin\left(\frac{x}{2} + \frac{\pi}{4}\right) = \sin\frac{x}{2}\cos\frac{\pi}{4} + \cos\frac{x}{2}\sin\frac{\pi}{4} \= \frac{\sqrt{2}}{2}(\sin\frac{x}{2} + \cos\frac{x}{2})\]and\[\cos\left(\frac{x}{2} + \frac{\pi}{4}\right) = \cos\frac{x}{2}\cos\frac{\pi}{4} - \sin\frac{x}{2}\sin\frac{\pi}{4} \= \frac{\sqrt{2}}{2}(\cos\frac{x}{2} - \sin\frac{x}{2})\].

Step 4: Simplify the tangent squared

Now, substitute these results back into the tangent formula:\[\tan^2\left(\frac{x}{2} + \frac{\pi}{4}\right) = \left(\frac{\sin\frac{x}{2} + \cos\frac{x}{2}}{\cos\frac{x}{2} - \sin\frac{x}{2}}\right)^2\]Simplifying this expression gives:\[\left( \frac{\sin\frac{x}{2} + \cos\frac{x}{2}}{\cos\frac{x}{2} - \sin\frac{x}{2}} \right)^2.\]

Step 5: Apply Pythagorean Identity

The Pythagorean identity \( \sin^2\left(\frac{x}{2}\right) + \cos^2\left(\frac{x}{2}\right) = 1 \) can be used to simplify further:\[\frac{(\sin\frac{x}{2} + \cos\frac{x}{2})^2}{(\cos\frac{x}{2} - \sin\frac{x}{2})^2} = \frac{1+\sin x}{1-\sin x}\]}

Step 6: Final Calculation Step

Now simplify and show that both sides are equal to verify the identity. Eventually you reach:\[\frac{(1 + \sin x)^2}{1 - \sin^2 x} = \frac{1+\sin x}{1 - \sin x}.\]And simplifying leads to confirming the identity as:\[\frac{1+\sin x}{1-\sin x},\]proving the identity is correct.

Key Concepts

Angle Sum Formulas

Let's explore the angle sum formulas, which are vital in trigonometry. These formulas enable us to find the sine or cosine of the sum of two angles. Specifically:

• The sine of a sum is given by \( \sin(a + b) = \sin a \cos b + \cos a \sin b \).
• The cosine of a sum is \( \cos(a + b) = \cos a \cos b - \sin a \sin b \).

Such formulas simplify computations and develop longer expressions into manageable parts. In our exercise, we applied these to \( \sin\left(\frac{x}{2} + \frac{\pi}{4}\right) \) and \( \cos\left(\frac{x}{2} + \frac{\pi}{4}\right) \) to write them as combinations of \( \sin\frac{x}{2}, \cos\frac{x}{2}, \sin\frac{\pi}{4}, \) and \( \cos\frac{\pi}{4} \).
By using known values for sine and cosine of \( \frac{\pi}{4} \), which is \( \frac{\sqrt{2}}{2} \), you replace complex expressions with simpler terms that are easy to manage. This approach is often essential in transforming complex trigonometric identities into simpler equations, aiding proofs and problem-solving.

Pythagorean Identity

The Pythagorean identity is a fundamental relationship in trigonometry that expresses the inherent link between sine and cosine for any angle. This identity is written as:

• \( \sin^2\theta + \cos^2\theta = 1 \)

It originates from the geometrical interpretation of the unit circle. If you draw a right-angled triangle within this circle, \( \sin\theta \) is the height, and \( \cos\theta \) is the base; the hypotenuse is the radius, which is 1.
This identity is pivotal in simplifying equations. In the exercise, it was utilized to further resolve the identity \( \tan^2\left(\frac{x}{2} + \frac{\pi}{4}\right) = \frac{1 + \sin x}{1 - \sin x} \). After expressing functions with angle sum formulas, our equation involves \( \sin^2\left(\frac{x}{2}\right) \) and \( \cos^2\left(\frac{x}{2}\right) \), which sum to one, making the simplification both possible and straightforward. Understanding this identity unlocks a range of problem-solving techniques.

Tangent Function

Tangent: it's more than just an angle ratio. This function, \( \tan\theta = \frac{\sin\theta}{\cos\theta} \), is defined based on sine and cosine. Because it relates these two functions, its behavior affects calculations in trigonometry profoundly.

• The function is undefined when \( \cos\theta = 0 \) because dividing by zero is mathematically undefined.
• \( \tan\theta \) shows periodic behavior, repeating every \( \pi \) radians.

In proving our identity, the tangent squared was the starting block. By square-forming \( \tan\left(\frac{x}{2} + \frac{\pi}{4}\right) \) using angle sum identities, we made it manageable by simplifying it in terms of sine and cosine, again showcasing the interconnectedness of trigonometric functions.
Understanding how to manipulate tangent within equations, and recognizing its relationship with sine and cosine, provides a strong advantage in accessing higher-level trigonometric proofs.