Substituting csc²θ in place of (1)/(sin²θ), the expression simplifies to cot²θ - csc²θ.

From the fundamental Pythagorean identity 1 + cot²θ = csc²θ, subtracting csc²θ and 1 from both sides gives cot²θ - csc²θ = -1.

3) $\cot^2\theta - \frac{1}{\sin^2\theta} = -1$

Answer: Hence proved, $\cot^2\theta - \frac{1}{\sin^2\theta} = -1$ .

Step-by-step solution

Step 1: State the Left Hand Side (LHS) and recall reciprocal identity

We begin with the left-hand side of the given identity. Recall the fundamental reciprocal identity that $\frac{1}{\sin\theta} = \csc\theta$ , which implies that $\frac{1}{\sin^2\theta} = \csc^2\theta$ .

Step 2: Substitute the reciprocal identity into the expression

Substituting $\csc^2\theta$ in place of $\frac{1}{\sin^2\theta}$ , the expression simplifies to $\cot^2\theta - \csc^2\theta$ .

Step 3: Apply the standard Pythagorean identity

From the fundamental Pythagorean identity $1 + \cot^2\theta = \csc^2\theta$ , subtracting $\csc^2\theta$ and $1$ from both sides gives $\cot^2\theta - \csc^2\theta = -1$ .

Step 4: Conclude the proof

The value of the left-hand side is $-1$ , which is equal to the right-hand side. Hence, the identity is proved.

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