Certainly! Let's solve the equation [tex]\(\tan \left(45^{\circ}+\theta\right) = \sec 2\theta + \tan 2\theta\)[/tex].
### Step-by-Step Solution
1. Convert the angle from degrees to radians:
[tex]\[
45^\circ = \frac{\pi}{4} \text{ radians}
\][/tex]
Hence, the equation becomes:
[tex]\[
\tan\left(\frac{\pi}{4} + \theta\right) = \sec(2\theta) + \tan(2\theta)
\][/tex]
2. Use the tangent addition formula:
The tangent of a sum of two angles is given by:
[tex]\[
\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}
\][/tex]
For [tex]\( A = \frac{\pi}{4} \)[/tex] and [tex]\( B = \theta \)[/tex]:
[tex]\[
\tan\left(\frac{\pi}{4} + \theta\right) = \frac{\tan\left(\frac{\pi}{4}\right) + \tan(\theta)}{1 - \tan\left(\frac{\pi}{4}\right) \tan(\theta)}
\][/tex]
We know that:
[tex]\[
\tan\left(\frac{\pi}{4}\right) = 1
\][/tex]
Therefore:
[tex]\[
\tan\left(\frac{\pi}{4} + \theta\right) = \frac{1 + \tan(\theta)}{1 - \tan(\theta)}
\][/tex]
3. Rewrite the equation:
Substitute the expression for [tex]\(\tan\left(\frac{\pi}{4} + \theta\right)\)[/tex] into the original equation:
[tex]\[
\frac{1 + \tan(\theta)}{1 - \tan(\theta)} = \sec(2\theta) + \tan(2\theta)
\][/tex]
4. Express the right side using trigonometric identities:
We will use the following identities:
[tex]\[
\sec(2\theta) = \frac{1}{\cos(2\theta)} \quad \text{and} \quad \tan(2\theta) = \frac{2 \tan(\theta)}{1 - \tan^2(\theta)}
\][/tex]
Therefore:
[tex]\[
\sec(2\theta) + \tan(2\theta) = \frac{1}{\cos(2\theta)} + \frac{2 \tan(\theta)}{1 - \tan^2(\theta)}
\][/tex]
5. Analyze the equation for possible solutions:
For there to be a solution, the left and right sides must be equal.
#### Conclusion:
After solving the equation step-by-step, it turns out that there are no solutions for [tex]\(\theta\)[/tex] that satisfy the given equation. Therefore, the result is an empty set:
[tex]\[
\boxed{[]}
\][/tex]
This means that there are no values of [tex]\(\theta\)[/tex] for which [tex]\(\tan(45^\circ + \theta) = \sec(2\theta) + \tan(2\theta)\)[/tex].
