Finding cotθ when cosecθ + cotθ is Given

The problem asks us to find the value of cotθ given the equation cosecθ + cotθ = 2. We can solve this using a fundamental trigonometric identity that relates cosecant and cotangent.

Using a Key Trigonometric Identity

There is a well-known Pythagorean identity involving cosecant and cotangent:

$\text{cosec}^2\theta - \text{cot}^2\theta = 1$

This identity is in the form of a difference of squares ($a^2 - b^2$), which can be factored as $(a-b)(a+b)$. Applying this to the identity, we get:

$(\text{cosec}\theta - \text{cot}\theta)(\text{cosec}\theta + \text{cot}\theta) = 1$

Substituting the Given Value

We are given that $\text{cosec}\theta + \text{cot}\theta = 2$. We can substitute this value into the factored identity:

$(\text{cosec}\theta - \text{cot}\theta)(2) = 1$

Now, we can solve for $(\text{cosec}\theta - \text{cot}\theta)$:

$\text{cosec}\theta - \text{cot}\theta = \frac{1}{2}$

$\text{cosec}\theta - \text{cot}\theta = 0.5$

Solving Simultaneous Equations

We now have two simple linear equations involving $\text{cosec}\theta$ and $\text{cot}\theta$:

1. $\text{cosec}\theta + \text{cot}\theta = 2$ (Given)
2. $\text{cosec}\theta - \text{cot}\theta = 0.5$ (Derived)

To find $\text{cot}\theta$, we can subtract the second equation from the first equation:

$(\text{cosec}\theta + \text{cot}\theta) - (\text{cosec}\theta - \text{cot}\theta) = 2 - 0.5$

Removing the parentheses and simplifying the left side:

$\text{cosec}\theta + \text{cot}\theta - \text{cosec}\theta + \text{cot}\theta = 1.5$

The $\text{cosec}\theta$ terms cancel out:

$2 \text{cot}\theta = 1.5$

Now, solve for $\text{cot}\theta$ by dividing both sides by 2:

$\text{cot}\theta = \frac{1.5}{2}$

$\text{cot}\theta = 0.75$

Conclusion

Based on the given information and the trigonometric identity, the value of $\text{cot}\theta$ is 0.75.

Revision Table: Key Steps

Additional Information on Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variable for which both sides of the equation are defined. They are crucial tools for simplifying expressions and solving trigonometric equations. The identity used here, $\text{cosec}^2\theta - \text{cot}^2\theta = 1$, is derived from the fundamental Pythagorean identity $\text{sin}^2\theta + \text{cos}^2\theta = 1$. Dividing the fundamental identity by $\text{sin}^2\theta$ (assuming $\text{sin}\theta \neq 0$) gives:

$\frac{\text{sin}^2\theta}{\text{sin}^2\theta} + \frac{\text{cos}^2\theta}{\text{sin}^2\theta} = \frac{1}{\text{sin}^2\theta}$

$1 + \text{cot}^2\theta = \text{cosec}^2\theta$

Rearranging this equation gives the identity used in the solution:

$\text{cosec}^2\theta - \text{cot}^2\theta = 1$

Understanding these identities allows us to manipulate trigonometric expressions effectively.