To solve the given trigonometric equation [tex]\(\frac{1-\sin 60^{\circ}}{\cos 60^{\circ}} = 2-\sqrt{3}\)[/tex], let's go through it step-by-step:
1. Identify the Trigonometric Values:
[tex]\[
\sin 60^{\circ} = \frac{\sqrt{3}}{2}
\][/tex]
[tex]\[
\cos 60^{\circ} = \frac{1}{2}
\][/tex]
2. Calculate the Numerator:
The numerator of the left-hand side is:
[tex]\[
1 - \sin 60^{\circ} = 1 - \frac{\sqrt{3}}{2}
\][/tex]
Simplifying this, we get:
[tex]\[
1 - \frac{\sqrt{3}}{2} \approx 0.1339745962155614
\][/tex]
3. Calculate the Denominator:
The denominator of the left-hand side is:
[tex]\[
\cos 60^{\circ} = \frac{1}{2} \approx 0.5
\][/tex]
4. Calculate the Left-hand Side:
The left-hand side of the equation now becomes:
[tex]\[
\frac{1 - \frac{\sqrt{3}}{2}}{\frac{1}{2}}
\][/tex]
Plugging in the approximate values we calculated:
[tex]\[
\frac{0.1339745962155614}{0.5} \approx 0.2679491924311228
\][/tex]
5. Evaluate the Right-hand Side:
The right-hand side of the equation is:
[tex]\[
2 - \sqrt{3} \approx 0.2679491924311228
\][/tex]
6. Compare Both Sides:
The left-hand side value we calculated is approximately [tex]\(0.2679491924311228\)[/tex], which matches the right-hand side value [tex]\(0.2679491924311228\)[/tex].
Since both sides are equal, the trigonometric equation [tex]\(\frac{1-\sin 60^{\circ}}{\cos 60^{\circ}} = 2-\sqrt{3}\)[/tex] holds true.
