Let's solve the expression [tex]\(\frac{\sin^2 A + 1}{\sin A \times \cos A}\)[/tex] step-by-step.
1. Identifying the Expression:
We start by carefully rewriting the given expression:
[tex]\[
\frac{\sin^2 A + 1}{\sin A \times \cos A}
\][/tex]
2. Breaking Down the Numerator:
The numerator of this fraction is [tex]\(\sin^2 A + 1\)[/tex].
3. Understanding the Structure:
The denominator consists of the product of [tex]\(\sin A\)[/tex] and [tex]\(\cos A\)[/tex].
4. Simplifying the Expression:
We will split the fraction into two separate fractions to simplify:
[tex]\[
\frac{\sin^2 A + 1}{\sin A \times \cos A} = \frac{\sin^2 A}{\sin A \times \cos A} + \frac{1}{\sin A \times \cos A}
\][/tex]
5. Simplifying Each Term Separately:
- For the first part [tex]\(\frac{\sin^2 A}{\sin A \times \cos A}\)[/tex]:
[tex]\[
\frac{\sin^2 A}{\sin A \times \cos A} = \frac{\sin A \times \sin A}{\sin A \times \cos A} = \frac{\sin A}{\cos A} = \tan A
\][/tex]
- For the second part [tex]\(\frac{1}{\sin A \times \cos A}\)[/tex]:
[tex]\[
\frac{1}{\sin A \times \cos A}
\][/tex]
6. Combining the Simplified Terms:
From the above steps, combining both parts gives:
[tex]\[
\tan A + \frac{1}{\sin A \times \cos A}
\][/tex]
This result simplifies further into trigonometric identities or remains expressed in the above format based on the requirement. However, since we initially stated the fraction form and each part was simplified clearly, we now know that:
[tex]\[
\frac{\sin^2 A + 1}{\sin A \times \cos A} = (sin(A)^2 + 1) / (sin(A)*cos(A))
\][/tex]
Thus, the fully simplified solution to [tex]\(\frac{\sin^2 A + 1}{\sin A \times \cos A}\)[/tex] is indeed [tex]\((sin(A)^2 + 1) / (sin(A)*cos(A))\)[/tex].
