Of course, let's solve this step-by-step.
Given the equation:
[tex]$\sqrt{\frac{3}{\sin 20^\circ}} - \frac{1}{\cos 20^\circ} = 4$[/tex]
We need to determine whether this equation holds true.
### Step-by-Step Solution:
1. Identify the trigonometric values:
- [tex]\(\sin 20^\circ\)[/tex] and [tex]\(\cos 20^\circ\)[/tex]
2. Calculate [tex]\(\sqrt{\frac{3}{\sin 20^\circ}}\)[/tex]:
First, find [tex]\(\frac{3}{\sin 20^\circ}\)[/tex], and then take the square root of the result.
3. Calculate [tex]\(\frac{1}{\cos 20^\circ}\)[/tex]:
This will be evaluated directly using the value of [tex]\(\cos 20^\circ\)[/tex].
4. Subtract [tex]\(\frac{1}{\cos 20^\circ}\)[/tex] from [tex]\(\sqrt{\frac{3}{\sin 20^\circ}}\)[/tex]:
Combine the results obtained from the above calculations.
5. Verify if the left-hand side equals 4:
Check if the final value equals 4.
### Detailed Calculation:
1. Evaluate [tex]\(\sin 20^\circ\)[/tex] and [tex]\(\cos 20^\circ\)[/tex]:
Let's find the approximate values:
- [tex]\(\sin 20^\circ \approx 0.3420\)[/tex]
- [tex]\(\cos 20^\circ \approx 0.9397\)[/tex]
2. Calculate [tex]\(\sqrt{\frac{3}{\sin 20^\circ}}\)[/tex]:
[tex]\[
\frac{3}{\sin 20^\circ} \approx \frac{3}{0.3420} \approx 8.7719
\][/tex]
[tex]\[
\sqrt{8.7719} \approx 2.9625
\][/tex]
3. Calculate [tex]\(\frac{1}{\cos 20^\circ}\)[/tex]:
[tex]\[
\frac{1}{\cos 20^\circ} \approx \frac{1}{0.9397} \approx 1.0642
\][/tex]
4. Subtract the values:
[tex]\[
\sqrt{\frac{3}{\sin 20^\circ}} - \frac{1}{\cos 20^\circ} \approx 2.9625 - 1.0642 \approx 1.8983
\][/tex]
### Conclusion:
The left side of the equation evaluates to approximately 1.8983, which is not equal to 4. Thus, given this detailed step-by-step analysis, the left-hand side does not equal the right-hand side as specified by the original equation.
