Solving Trigonometric Equations: Finding cos 2θ

The problem asks us to find the value of \( \cos 2\theta \) given the equation \( \tan(45^\circ + \theta) = 1 + \sin 2\theta \), with the constraint \( -\frac{\pi}{4} < \theta < \frac{\pi}{4} \).

Analyzing the Given Trigonometric Equation

We are given the equation:

\( \tan(45^\circ + \theta) = 1 + \sin 2\theta \)

We need to simplify both sides of the equation using standard trigonometric identities.

Applying Trigonometric Identities

First, let's simplify the left side using the angle addition formula for tangent: \( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \).

Here, \( A = 45^\circ \) and \( B = \theta \). We know that \( \tan 45^\circ = 1 \).

So, \( \tan(45^\circ + \theta) = \frac{\tan 45^\circ + \tan \theta}{1 - \tan 45^\circ \tan \theta} = \frac{1 + \tan \theta}{1 - \tan \theta} \).

Now, let's express the right side, \( 1 + \sin 2\theta \), in terms of \( \tan \theta \). We use the identity \( \sin 2\theta = \frac{2 \tan \theta}{1 + \tan^2 \theta} \).

So, \( 1 + \sin 2\theta = 1 + \frac{2 \tan \theta}{1 + \tan^2 \theta} \).

Now, substitute these simplified expressions back into the original equation:

\( \frac{1 + \tan \theta}{1 - \tan \theta} = 1 + \frac{2 \tan \theta}{1 + \tan^2 \theta} \)

Solving the Equation for tan θ

Let \( t = \tan \theta \) to make the algebra simpler. The equation becomes:

\( \frac{1 + t}{1 - t} = 1 + \frac{2t}{1 + t^2} \)

Simplify the right side by finding a common denominator:

\( 1 + \frac{2t}{1 + t^2} = \frac{1(1 + t^2)}{1 + t^2} + \frac{2t}{1 + t^2} = \frac{1 + t^2 + 2t}{1 + t^2} = \frac{(1 + t)^2}{1 + t^2} \)

So, the equation is:

\( \frac{1 + t}{1 - t} = \frac{(1 + t)^2}{1 + t^2} \)

Cross-multiply:

\( (1 + t)(1 + t^2) = (1 - t)(1 + t)^2 \)

\( (1 + t)(1 + t^2) = (1 - t)(1 + 2t + t^2) \)

Expand both sides:

\( 1 + t^2 + t + t^3 = 1(1 + 2t + t^2) - t(1 + 2t + t^2) \)

\( 1 + t + t^2 + t^3 = 1 + 2t + t^2 - t - 2t^2 - t^3 \)

\( 1 + t + t^2 + t^3 = 1 + t - t^2 - t^3 \)

Move all terms to one side:

\( (1 + t + t^2 + t^3) - (1 + t - t^2 - t^3) = 0 \)

\( 1 + t + t^2 + t^3 - 1 - t + t^2 + t^3 = 0 \)

\( 2t^2 + 2t^3 = 0 \)

Factor out \( 2t^2 \):

\( 2t^2(1 + t) = 0 \)

This equation gives two possible solutions for \( t \):

1. \( 2t^2 = 0 \implies t^2 = 0 \implies t = 0 \)

2. \( 1 + t = 0 \implies t = -1 \)

Since \( t = \tan \theta \), we have \( \tan \theta = 0 \) or \( \tan \theta = -1 \).

Considering the Given Range for θ

The problem states that \( -\frac{\pi}{4} < \theta < \frac{\pi}{4} \). We need to check which of the values for \( \tan \theta \) are valid within this range.

Recall that \( \tan \theta = 0 \) when \( \theta \) is an integer multiple of \( \pi \) (or \( 180^\circ \)). The value \( \theta = 0 \) is within the range \( -\frac{\pi}{4} < \theta < \frac{\pi}{4} \) (which corresponds to \( -45^\circ < \theta < 45^\circ \)).

Recall that \( \tan \theta = -1 \) when \( \theta = \frac{3\pi}{4} + n\pi \) or \( \theta = -\frac{\pi}{4} + n\pi \) for integer \( n \). For \( n = 0 \), \( \theta = -\frac{\pi}{4} \). However, the given range is \( -\frac{\pi}{4} < \theta < \frac{\pi}{4} \), which uses strict inequalities. This means \( \theta = -\frac{\pi}{4} \) is not included in the allowed range.

Therefore, the only valid solution for \( \theta \) in the given range is \( \theta = 0 \), which corresponds to \( \tan \theta = 0 \).

Calculating the Value of cos 2θ

We found that the valid value for \( \theta \) is \( 0 \).

Now we need to calculate \( \cos 2\theta \).

Substitute \( \theta = 0 \) into the expression \( \cos 2\theta \):

\( \cos(2 \times 0) = \cos(0) \)

The value of \( \cos(0) \) is \( 1 \).

So, the value of \( \cos 2\theta \) is \( 1 \).

Summary of Steps

Here is a summary of the steps taken to solve the problem:

1. Analyze the given equation \( \tan(45^\circ + \theta) = 1 + \sin 2\theta \).
2. Apply the angle addition formula for tangent: \( \tan(45^\circ + \theta) = \frac{1 + \tan \theta}{1 - \tan \theta} \).
3. Apply the double angle formula for sine in terms of tangent: \( \sin 2\theta = \frac{2 \tan \theta}{1 + \tan^2 \theta} \).
4. Substitute these expressions back into the equation.
5. Solve the resulting algebraic equation for \( t = \tan \theta \).
6. Identify the possible values for \( \tan \theta \).
7. Use the given range \( -\frac{\pi}{4} < \theta < \frac{\pi}{4} \) to determine the valid value(s) of \( \theta \).
8. Calculate \( \cos 2\theta \) using the valid value of \( \theta \).

The final answer is \( 1 \).

Revision Table: Key Trigonometry Concepts

Additional Information: Exploring Trigonometric Identities

Trigonometric identities are equalities that involve trigonometric functions and are true for every single value of the occurring variables for which both sides of the equality are defined. They are fundamental in simplifying expressions and solving trigonometric equations.

Besides the identities used in this problem, there are many others, such as:

• Pythagorean Identities (e.g., \( \sin^2 \theta + \cos^2 \theta = 1 \))
• Reciprocal Identities (e.g., \( \sec \theta = \frac{1}{\cos \theta} \))
• Quotient Identities (e.g., \( \tan \theta = \frac{\sin \theta}{\cos \theta} \))
• Half Angle Formulas
• Sum-to-Product and Product-to-Sum Formulas

Understanding these identities and knowing when to apply them is essential for solving a wide range of trigonometry problems. Practice is key to mastering the application of these formulas.