If $\sin x = \frac{4}{5}$, find $\frac{1 + \cot^2 x}{\csc^2 x - 1}$.

1. Analyze the question

We are given $\sin x = \frac{4}{5}$ and asked to evaluate the expression $\frac{1 + \cot^2 x}{\csc^2 x - 1}$.

2. Explain the concept

To solve this, we use fundamental trigonometric identities:

• Pythagorean Identity: $1 + \cot^2 x = \csc^2 x$
• Rearranged Identity: $\csc^2 x - 1 = \cot^2 x$

By substituting these identities into the expression, we can simplify the problem significantly before plugging in the value of $\sin x$.

3. Show the solution

Step 1: Simplify the expression using identities. The expression is:

Using the identity $1 + \cot^2 x = \csc^2 x$, the numerator becomes $\csc^2 x$. Using the identity $\csc^2 x - 1 = \cot^2 x$, the denominator becomes $\cot^2 x$.

So, the expression simplifies to:

Step 2: Express in terms of $\sin x$ and $\cos x$. Recall that $\csc x = \frac{1}{\sin x}$ and $\cot x = \frac{\cos x}{\sin x}$.

Step 3: Find $\cos x$ and solve. Given $\sin x = \frac{4}{5}$, we use the identity $\sin^2 x + \cos^2 x = 1$:

Now, substitute $\cos^2 x$ into our simplified expression ($\frac{1}{\cos^2 x}$):

4. Identify the answer

Comparing our result $\frac{25}{9}$ with the given options, it matches option B.

5. Tip

Whenever you see trigonometric expressions involving squares like $\csc^2 x$ or $\cot^2 x$, always check if you can simplify them using the Pythagorean identities ($1 + \tan^2 x = \sec^2 x$ or $1 + \cot^2 x = \csc^2 x$) before substituting the numerical values. It saves time and reduces the chance of calculation errors!