Sure, I'll walk you through the detailed, step-by-step solution for evaluating [tex]\(\tan A + \frac{1}{\sin A \cdot \cos A}\)[/tex].
1. Understand the Problem:
We are asked to evaluate the expression [tex]\(\tan A + \frac{1}{\sin A \cos A}\)[/tex].
2. Separate the Expression:
Let's break down the expression into two parts:
[tex]\[
\tan A + \frac{1}{\sin A \cos A}
\][/tex]
3. Evaluate [tex]\(\tan A\)[/tex]:
Recall the trigonometric identity for the tangent of an angle [tex]\(A\)[/tex]:
[tex]\[
\tan A = \frac{\sin A}{\cos A}
\][/tex]
4. Evaluate [tex]\(\frac{1}{\sin A \cos A}\)[/tex]:
Notice that [tex]\(\sin A \cos A\)[/tex] can be written using the double-angle identity for sine:
[tex]\[
\sin 2A = 2 \sin A \cos A \implies \sin A \cos A = \frac{\sin 2A}{2}
\][/tex]
Thus:
[tex]\[
\frac{1}{\sin A \cos A} = \frac{1}{\frac{\sin 2A}{2}} = \frac{2}{\sin 2A}
\][/tex]
5. Add the Two Terms:
Now, we combine the results from steps 3 and 4:
[tex]\[
\tan A + \frac{2}{\sin 2A}
\][/tex]
For the given specific values:
6. Specific Angle in Radians (A):
Let's consider [tex]\(A = \frac{\pi}{4}\)[/tex] (which is 45 degrees), based on our problem:
- [tex]\(\tan A = \tan \left(\frac{\pi}{4}\right) = 1\)[/tex]
- For [tex]\(A = \frac{\pi}{4}\)[/tex]:
[tex]\[
\sin A = \sin \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}
\][/tex]
[tex]\[
\cos A = \cos \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}
\][/tex]
Thus:
[tex]\[
\sin A \cos A = \left(\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{2}}{2}\right) = \frac{2}{4} = \frac{1}{2}
\][/tex]
Therefore:
[tex]\[
\frac{1}{\sin A \cos A} = \frac{1}{\frac{1}{2}} = 2
\][/tex]
7. Calculate the Final Result:
Now sum the two results:
[tex]\[
\tan A + \frac{1}{\sin A \cos A} = 1 + 2 = 3
\][/tex]
So the detailed, step-by-step evaluated result of the expression is:
[tex]\[
\tan A + \frac{1}{\sin A \cos A} = 3
\][/tex]
Additionally, breaking down the specific values for interest:
- [tex]\(\tan A \approx 0.9999999999999999\)[/tex]
- [tex]\( \frac{1}{\sin A \cos A} = 2.0 \)[/tex]
And summing them:
[tex]\[
0.9999999999999999 + 2.0 = 3.0
\][/tex]
