Question

For an acute angle $\theta$, if $\sin \theta = \frac{1}{9}$, then value of $\frac{9\csc\theta + 1}{9\csc\theta - 1}$ is

• A. $0$
• B. $\frac{80}{81}$
• C. $1$
• D. $\frac{82}{80}$

Hint:

Always convert complex trigonometric expressions into their basic counterparts.
Since $\csc\theta = \frac{1}{\sin\theta}$, substituting this directly yields $\frac{\frac{9}{\sin\theta} + 1}{\frac{9}{\sin\theta} - 1} = \frac{9 + \sin\theta}{9 - \sin\theta}$, which is extremely quick to compute!

Answer 1

The Correct Option is D

Step 1: Rewrite the expression purely in terms of $\sin\theta$.
Since $\csc\theta = \frac{1}{\sin\theta}$, we have $9\csc\theta = \frac{9}{\sin\theta}$.
Step 2: Substitute $\sin\theta = \frac{1}{9}$ directly.
\[ \frac{9}{\sin\theta} = 9 \div \frac{1}{9} = 81 \]
Step 3: Plug this value of $9\csc\theta$ into the given fraction.
\[ \frac{81+1}{81-1} = \frac{82}{80} \]
Step 4: Conclude.
This matches option (4).
\[ \boxed{\frac{82}{80}} \]