Question

1. \\(\frac{(\csc^{2}\theta - 1)}{\csc^{2}\theta}\\)

Explanation:

Step1: Recall Pythagorean identity

We know the Pythagorean identity for cosecant and cotangent: $\csc^{2}\theta - 1=\cot^{2}\theta$. Substitute this into the numerator.
$\frac{\cot^{2}\theta}{\csc^{2}\theta}$

Step2: Express in terms of sine and cosine

Recall that $\cot\theta=\frac{\cos\theta}{\sin\theta}$ and $\csc\theta = \frac{1}{\sin\theta}$. So $\cot^{2}\theta=\frac{\cos^{2}\theta}{\sin^{2}\theta}$ and $\csc^{2}\theta=\frac{1}{\sin^{2}\theta}$. Substitute these into the fraction:
$\frac{\frac{\cos^{2}\theta}{\sin^{2}\theta}}{\frac{1}{\sin^{2}\theta}}$

Step3: Simplify the fraction

Dividing by a fraction is multiplying by its reciprocal, so $\frac{\cos^{2}\theta}{\sin^{2}\theta}\times\sin^{2}\theta=\cos^{2}\theta$. Alternatively, we can also use the identity $\frac{\cot^{2}\theta}{\csc^{2}\theta}=\cos^{2}\theta$ (since $\cot\theta=\frac{\cos\theta}{\sin\theta}$ and $\csc\theta=\frac{1}{\sin\theta}$, so the $\sin^{2}\theta$ terms cancel out). Another way: we know that $1 - \sin^{2}\theta=\cos^{2}\theta$, and since $\csc^{2}\theta=\frac{1}{\sin^{2}\theta}$, we can also think of $\frac{\csc^{2}\theta - 1}{\csc^{2}\theta}=1-\frac{1}{\csc^{2}\theta}=1 - \sin^{2}\theta=\cos^{2}\theta$.

Answer:

$\cos^{2}\theta$