https://file.aiquickdraw.com/mathbot/file/17269023475986qvrx1z5.jpg

The equation in the image is:

\[
\text{Show that } \cot 2\theta + \tan \theta = \csc 2\theta.
\]

Let's prove this step by step using trigonometric identities.

### Step 1: Express \(\cot 2\theta\) and \(\csc 2\theta\) in terms of basic trigonometric functions
We know the identities:
\[
\cot 2\theta = \frac{\cos 2\theta}{\sin 2\theta},
\]
\[
\csc 2\theta = \frac{1}{\sin 2\theta}.
\]

### Step 2: Use the identity for \(\tan \theta\)
\[
\tan \theta = \frac{\sin \theta}{\cos \theta}.
\]

### Step 3: Combine these expressions

Now, substitute these into the left-hand side of the equation:

\[
\cot 2\theta + \tan \theta = \frac{\cos 2\theta}{\sin 2\theta} + \frac{\sin \theta}{\cos \theta}.
\]

### Step 4: Get a common denominator for the terms

The common denominator of \(\sin 2\theta\) and \(\cos \theta\) is \(\sin 2\theta \cdot \cos \theta\). So, rewrite the terms with this common denominator:

\[
\frac{\cos 2\theta}{\sin 2\theta} + \frac{\sin \theta}{\cos \theta} = \frac{\cos 2\theta \cdot \cos \theta + \sin \theta \cdot \sin 2\theta}{\sin 2\theta \cdot \cos \theta}.
\]

### Step 5: Simplify the numerator

Use the double angle identity for sine, \(\sin 2\theta = 2\sin \theta \cos \theta\), in the numerator:

\[
\cos 2\theta \cdot \cos \theta + \sin \theta \cdot 2 \sin \theta \cos \theta = \cos 2\theta \cdot \cos \theta + 2 \sin^2 \theta \cdot \cos \theta.
\]

Factor out \(\cos \theta\) from the numerator:

\[
\cos \theta (\cos 2\theta + 2 \sin^2 \theta).
\]

### Step 6: Simplify using the Pythagorean identity

Using the identity \(\sin^2 \theta + \cos^2 \theta = 1\), rewrite \(\cos 2\theta\) as:

\[
\cos 2\theta = \cos^2 \theta - \sin^2 \theta.
\]

So,

\[
\cos 2\theta + 2 \sin^2 \theta = (\cos^2 \theta - \sin^2 \theta) + 2 \sin^2 \theta = \cos^2 \theta + \sin^2 \theta = 1.
\]

Thus, the numerator becomes:

\[
\cos \theta \cdot 1 = \cos \theta.
\]

### Step 7: Final simplification

Now, the expression simplifies to:

\[
\frac{\cos \theta}{\sin 2\theta \cdot \cos \theta}.
\]

Cancel out \(\cos \theta\) from the numerator and denominator:

\[
\frac{1}{\sin 2\theta}.
\]

This is equal to \(\csc 2\theta\), which is the right-hand side of the equation.

Thus, we have shown that:

\[
\cot 2\theta + \tan \theta = \csc 2\theta.
\]

Would you like further clarification on any steps or additional details?

---

Here are 5 related questions to extend your understanding:
1. What are the fundamental trigonometric identities used in this proof?
2. How can you derive the double angle formulas for sine and cosine?
3. Can you prove the identity \(\sin^2 \theta + \cos^2 \theta = 1\) geometrically?
4. How does the cotangent function relate to the tangent function?
5. What are other examples where cotangent and cosecant functions appear in trigonometric proofs?

**Tip:** When dealing with trigonometric identities, it's often helpful to express all terms in terms of sine and cosine to simplify the equations.

A step-by-step guide to proving the trigonometric identity cot(2θ) + tan(θ) = csc(2θ). Learn about key concepts like trigonometric identities, the double angle formulas, and the Pythagorean identity, suitable for students in Grades 11-12.

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