Certainly! Let's solve the problem step by step:
1. Given Information:
We are given that [tex]\( \cot A = \sqrt{3} \)[/tex].
2. Determine [tex]\( \tan A \)[/tex]:
- We know the relation between cotangent and tangent:
[tex]\[
\cot A = \frac{1}{\tan A}
\][/tex]
- Given [tex]\( \cot A = \sqrt{3} \)[/tex], we can find [tex]\( \tan A \)[/tex]:
[tex]\[
\tan A = \frac{1}{\sqrt{3}} \approx 0.577
\][/tex]
3. Determine [tex]\( \sec^2 A \)[/tex]:
- The identity involving secant and tangent is:
[tex]\[
\sec^2 A = 1 + \tan^2 A
\][/tex]
- We know [tex]\( \tan A = 0.577 \)[/tex]:
[tex]\[
\tan^2 A = (0.577)^2 \approx 0.333
\][/tex]
- So:
[tex]\[
\sec^2 A = 1 + 0.333 \approx 1.333
\][/tex]
4. Determine [tex]\( \cos^2 A \)[/tex]:
- The relation between secant and cosine is:
[tex]\[
\sec A = \frac{1}{\cos A}
\][/tex]
- Therefore:
[tex]\[
\sec^2 A = \frac{1}{\cos^2 A}
\][/tex]
- Given that [tex]\( \sec^2 A \approx 1.333 \)[/tex]:
[tex]\[
\cos^2 A = \frac{1}{1.333} \approx 0.75
\][/tex]
5. Calculate [tex]\( \sec^2 A - \cos^2 A \)[/tex]:
- Now, we subtract [tex]\( \cos^2 A \)[/tex] from [tex]\( \sec^2 A \)[/tex]:
[tex]\[
\sec^2 A - \cos^2 A = 1.333 - 0.75 \approx 0.583
\][/tex]
Therefore, the value of [tex]\( \sec^2 A - \cos^2 A \)[/tex] is approximately [tex]\( 0.583 \)[/tex].
