	If tan⁡〖θ+1/tan⁡θ =2〗 then the value of cosec θ is
	1                     (b) 1/√2                     (c) √2                     (d) √3/2


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) \(\sqrt{2}\)**.

---

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.

If tan⁡(θ) + 1/tan⁡(θ) = 2, then the value of cosec θ is: (a) 1, (b) 1/√2, (c) √2, (d) √3/2

Learn how to solve a trigonometric problem where tan(θ) + 1/tan(θ) = 2. This solution involves using trigonometric identities and solving for cosec θ step-by-step. Suitable for high school students studying advanced trigonometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation