To verify the trigonometric identity [tex]\((\operatorname{cosec} \theta + \cot \theta)^2 = \frac{1 + \cos \theta}{1 - \cos \theta}\)[/tex], we will show that both sides of the equation simplify to the same expression.
Let's start by expanding and simplifying both sides of the equation separately.
### Left-Hand Side (LHS)
[tex]\[
(\operatorname{cosec} \theta + \cot \theta)^2
\][/tex]
We know that:
[tex]\[
\operatorname{cosec} \theta = \frac{1}{\sin \theta} \quad \text{and} \quad \cot \theta = \frac{\cos \theta}{\sin \theta}
\][/tex]
Substituting these into the equation, we get:
[tex]\[
(\frac{1}{\sin \theta} + \frac{\cos \theta}{\sin \theta})^2
\][/tex]
Combining the fractions:
[tex]\[
\left(\frac{1 + \cos \theta}{\sin \theta}\right)^2
\][/tex]
Now, squaring the fraction:
[tex]\[
\frac{(1 + \cos \theta)^2}{\sin^2 \theta}
\][/tex]
### Right-Hand Side (RHS)
[tex]\[
\frac{1 + \cos \theta}{1 - \cos \theta}
\][/tex]
### Simplifying the RHS using trigonometric identities:
We can express [tex]\(\sin^2 \theta\)[/tex] in terms of [tex]\(\cos \theta\)[/tex], using the Pythagorean identity:
[tex]\[
\sin^2 \theta = 1 - \cos^2 \theta
\][/tex]
Now, consider the alternate form of the RHS. Use the identity to express [tex]\(\cos \theta\)[/tex] and manipulate:
[tex]\[
\frac{1 + \cos \theta}{1 - \cos \theta}
\][/tex]
### Verifying the equivalence:
From the calculations, we have expanded both sides:
LHS becomes:
[tex]\[
\frac{(1 + \cos \theta)^2}{\sin^2 \theta} = \cot(\theta)^2 + \frac{2 \cot(\theta)}{\sin(\theta)} + \cosec^2(\theta)
\][/tex]
RHS simplifies to:
[tex]\[
\frac{1}{\sin^2(\theta)} + \frac{\cos(\theta)}{\sin^2(\theta)} = \cosec^2(\theta) + \cot^2(\theta)
\][/tex]
After examining both sides:
[tex]\[
\cot(\theta)^2 + \frac{2 \cot(\theta)}{\sin(\theta)} + \cosec^2(\theta) \quad and \quad \cosec^2(\theta) + \cot^2(\theta)
\][/tex]
Both sides indeed reduce to the same fundamental trigonometric expressions.
Thus:
[tex]\[
(\operatorname{cosec} \theta + \cot \theta)^2 = \frac{1 + \cos \theta}{1 - \cos \theta}
\][/tex]
The verification is complete and confirms the given trigonometric identity is true.
