$\begin{aligned}
(-4) & 1-\tan \theta \tan \frac{\pi}{4} \\
= & \frac{\left(-\frac{4 \sqrt{65}}{65}\right)+(1)}{1-\left(-\frac{4 \sqrt{65}}{65}\right)(1)}
\end{aligned}$
Which of the following is the correct answer?
A. $\frac{65+4 \sqrt{65}}{65-4 \sqrt{65}}$
B. $\frac{4-\sqrt{65}}{4+\sqrt{65}}$
C. $\frac{65-4 \sqrt{65}}{65+4 \sqrt{65}}$
D. $\frac{4+\sqrt{65}}{4-\sqrt{65}}$
Answer (1)
Simplify the given expression by multiplying the numerator and denominator by 65: \frac{\left(-\frac{4 \sqrt{65}}{65}\\right)+(1)}{1-\left(-\frac{4 \sqrt{65}}{65}\\right)(1)} = \frac{65 - 4 \sqrt{65}}{65 + 4 \sqrt{65}} .
Compare the simplified expression with the given options.
Identify the correct option as C.
The correct answer is 65 + 4 65 65 − 4 65
Explanation
Understanding the Problem The problem provides an equation with a trigonometric expression and asks us to simplify the right-hand side and choose the correct answer from the given options. The right-hand side is a fraction involving a square root. We need to simplify this fraction and compare it with the given options to find the correct match.
Writing the Expression The given expression is: \frac{\left(-\frac{4 \sqrt{65}}{65}\\\right)+(1)}{1-\left(-\frac{4 \sqrt{65}}{65}\\right)(1)}
Simplifying the Expression To simplify the expression, we multiply both the numerator and the denominator by 65: 65 × ( 1 − ( − 65 4 65 ) ) 65 × ( − 65 4 65 + 1 ) = 65 + 4 65 − 4 65 + 65 So, the simplified expression is: 65 + 4 65 65 − 4 65
Comparing with Options Now, we compare the simplified expression with the given options: A. 65 − 4 65 65 + 4 65 B. 4 + 65 4 − 65 C. 65 + 4 65 65 − 4 65 D. 4 − 65 4 + 65
We can see that option C matches our simplified expression.
Final Answer Therefore, the correct answer is: 65 + 4 65 65 − 4 65 which corresponds to option C.
Examples
This type of simplification is useful in various fields like physics and engineering when dealing with complex equations involving trigonometric functions and square roots. For example, when calculating impedance in electrical circuits or analyzing wave interference patterns, simplifying expressions can make the calculations easier and more manageable. Imagine you are designing a bridge and need to calculate the tension in a cable. The equation might involve square roots and fractions, and simplifying it will help you determine the precise amount of tension the cable needs to withstand, ensuring the bridge's safety.
