If cosec θ - cot θ = m, then what is cosec θ equal to?

Question

If cosec &theta; - cot &theta; = m, then what is cosec &theta; equal to?

Prepp Team · Accepted Answer

Solving Trigonometric Equations: Finding Cosec Theta

This problem asks us to find the value of $\text{cosec } \theta$ given a relationship between $\text{cosec } \theta$ and $\text{cot } \theta$. We are given the equation:

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

To solve this, we need to use a fundamental trigonometric identity that relates $\text{cosec } \theta$ and $\text{cot } \theta$. The relevant identity is:

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

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

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

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

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

Now, we can solve for the sum $\text{cosec } \theta + \text{cot } \theta$:

$\text{cosec } \theta + \text{cot } \theta = \frac{1}{m}$

Now we have a system of two simple linear equations involving $\text{cosec } \theta$ and $\text{cot } \theta$:

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

To find $\text{cosec } \theta$, we can add these two equations together. Notice that the $\text{cot } \theta$ terms will cancel out:

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

Combining like terms:

$2 \text{cosec } \theta = m + \frac{1}{m}$

Finally, to find $\text{cosec } \theta$, we divide both sides by 2:

$\text{cosec } \theta = \frac{1}{2} \left(m + \frac{1}{m}\right)$

Distributing the $\frac{1}{2}$ gives:

$\text{cosec } \theta = \frac{m}{2} + \frac{1}{2m}$

This expression gives $\text{cosec } \theta$ in terms of $m$. We can compare this result with the given options.

The calculated value $\frac{m}{2} + \frac{1}{2m}$ matches one of the options provided.

Revision Table: Key Trigonometric Identities

Understanding fundamental trigonometric identities is essential for solving problems like this one. Here's a table of some important identities:

Identity Type	Identity
Pythagorean Identity	$\sin^2 \theta + \cos^2 \theta = 1$
Pythagorean Identity	$1 + \tan^2 \theta = \sec^2 \theta$
Pythagorean Identity	$1 + \cot^2 \theta = \text{cosec}^2 \theta$
Reciprocal Identity	$\text{cosec } \theta = \frac{1}{\sin \theta}$
Reciprocal Identity	$\sec \theta = \frac{1}{\cos \theta}$
Reciprocal Identity	$\cot \theta = \frac{1}{\tan \theta}$
Quotient Identity	$\tan \theta = \frac{\sin \theta}{\cos \theta}$
Quotient Identity	$\cot \theta = \frac{\cos \theta}{\sin \theta}$

The identity $\text{cosec}^2 \theta - \text{cot}^2 \theta = 1$ used in this problem is a rearrangement of the identity $1 + \cot^2 \theta = \text{cosec}^2 \theta$.

Additional Information: Working with Cosec and Cot

The cosecant ($\text{cosec } \theta$) and cotangent ($\text{cot } \theta$) functions are reciprocal functions related to sine ($\sin \theta$) and tangent ($\tan \theta$), respectively. Specifically:

$\text{cosec } \theta = \frac{1}{\sin \theta}$ (defined when $\sin \theta \neq 0$)
$\text{cot } \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}$ (defined when $\sin \theta \neq 0$)

Problems involving $\text{cosec } \theta$ and $\text{cot } \theta$ often utilize the identity $\text{cosec}^2 \theta - \text{cot}^2 \theta = 1$. Recognizing this identity as a difference of squares is a common technique to derive useful relationships, as demonstrated in this solution.

When you are given an equation like $\text{cosec } \theta - \text{cot } \theta = m$, and you know the identity $\text{cosec}^2 \theta - \text{cot}^2 \theta = 1$, you can often find a second relationship, $\text{cosec } \theta + \text{cot } \theta = \frac{1}{m}$, by using the difference of squares factorization. This technique is very powerful for solving such problems.

Once you have the two linear equations:

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

You can solve for either $\text{cosec } \theta$ or $\text{cot } \theta$ by adding or subtracting the equations. Adding them eliminates $\text{cot } \theta$ and subtracting the first from the second eliminates $\text{cosec } \theta$.

Prepp Team · Answer

$m + \frac{1}{m}$

Prepp Team · Answer

$m - \frac{1}{m}$

Prepp Team · Answer

$\frac{m}{2} + \frac{2}{m}$

Solving Trigonometric Equations: Finding Cosec Theta

Revision Table: Key Trigonometric Identities

Additional Information: Working with Cosec and Cot

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Identity Type	Identity
Pythagorean Identity	\(\sin^2 \theta + \cos^2 \theta = 1\)
Pythagorean Identity	\(1 + \tan^2 \theta = \sec^2 \theta\)
Pythagorean Identity	\(1 + \cot^2 \theta = \text{cosec}^2 \theta\)
Reciprocal Identity	\(\text{cosec } \theta = \frac{1}{\sin \theta}\)
Reciprocal Identity	\(\sec \theta = \frac{1}{\cos \theta}\)
Reciprocal Identity	\(\cot \theta = \frac{1}{\tan \theta}\)
Quotient Identity	\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
Quotient Identity	\(\cot \theta = \frac{\cos \theta}{\sin \theta}\)