To solve for $\csc \theta$ given the equation $\tan \theta + \frac{1}{\tan \theta} = 2$, we can proceed as follows:

Step 1: Let $\tan \theta = x$

Given that: $\tan \theta + \frac{1}{\tan \theta} = 2$ Substitute $\tan \theta = x$: $x + \frac{1}{x} = 2$

Step 2: Solve the equation

Multiply both sides by $x$: $x^2 + 1 = 2x$ Rearranging terms gives: $x^2 - 2x + 1 = 0$ $(x - 1)^2 = 0$ So, $x = 1$.

Step 3: Find $\sin \theta$ and $\cos \theta$

Since $\tan \theta = 1$, we know that: $\sin \theta = \cos \theta$

Step 4: Find $\sin \theta$

We use the identity: $\tan \theta = \frac{\sin \theta}{\cos \theta} = 1 \implies \sin \theta = \cos \theta$

Since $\sin^2 \theta + \cos^2 \theta = 1$: $\sin^2 \theta + \sin^2 \theta = 1$ $2 \sin^2 \theta = 1$ $\sin^2 \theta = \frac{1}{2}$ $\sin \theta = \frac{1}{\sqrt{2}}$

Step 5: Find $\csc \theta$

The cosecant is the reciprocal of the sine: $\csc \theta = \frac{1}{\sin \theta} = \sqrt{2}$

Final Answer:

$\csc \theta = \sqrt{2}$

So, the correct option is (c) $\sqrt{2}$.

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Would you like a more detailed explanation or have any questions? Here are some related questions you might find interesting:

1. What is the relationship between $\tan \theta$ and $\csc \theta$?
2. How to solve equations involving trigonometric identities?
3. What is the value of $\sin \theta$ and $\cos \theta$ for $\theta = 45^\circ$?
4. How to derive the identity $\sin^2 \theta + \cos^2 \theta = 1$?
5. What is the reciprocal of $\cos \theta$ in trigonometry?

Tip: In trigonometry, knowing a few fundamental identities can simplify complex expressions and make it easier to solve equations.