Sure! To solve the expression [tex]\(\frac{1 + \tan^2 A}{1 + \cot^2 A}\)[/tex], let's break it down step-by-step using trigonometric identities.
### Step 1: Understand the trigonometric identities involved
1. Tangent and Cotangent:
- [tex]\(\tan A = \frac{\sin A}{\cos A}\)[/tex]
- [tex]\(\cot A = \frac{\cos A}{\sin A}\)[/tex]
2. Pythagorean Identity for Tangent:
- [tex]\(1 + \tan^2 A = \sec^2 A\)[/tex]
3. Pythagorean Identity for Cotangent:
- [tex]\(1 + \cot^2 A = \csc^2 A\)[/tex]
### Step 2: Substitute the identities
Replace [tex]\(1 + \tan^2 A\)[/tex] and [tex]\(1 + \cot^2 A\)[/tex] with their equivalent Pythagorean identities:
[tex]\[1 + \tan^2 A = \sec^2 A\][/tex]
[tex]\[1 + \cot^2 A = \csc^2 A\][/tex]
So, the expression now becomes:
[tex]\[\frac{\sec^2 A}{\csc^2 A}\][/tex]
### Step 3: Express secant and cosecant in terms of sine and cosine
Recall the definitions of secant and cosecant:
- [tex]\(\sec A = \frac{1}{\cos A}\)[/tex]
- [tex]\(\csc A = \frac{1}{\sin A}\)[/tex]
Therefore:
[tex]\[\sec^2 A = \left(\frac{1}{\cos A}\right)^2 = \frac{1}{\cos^2 A}\][/tex]
[tex]\[\csc^2 A = \left(\frac{1}{\sin A}\right)^2 = \frac{1}{\sin^2 A}\][/tex]
### Step 4: Substitute these into the expression
Now substituting [tex]\(\sec^2 A\)[/tex] and [tex]\(\csc^2 A\)[/tex] into the expression, we get:
[tex]\[
\frac{\frac{1}{\cos^2 A}}{\frac{1}{\sin^2 A}} = \frac{1 / \cos^2 A}{1 / \sin^2 A}
\][/tex]
### Step 5: Simplify the expression
Dividing by a fraction is the same as multiplying by its reciprocal:
[tex]\[
\frac{1 / \cos^2 A}{1 / \sin^2 A} = \left(\frac{1}{\cos^2 A}\right) \times \left(\frac{\sin^2 A}{1}\right) = \frac{\sin^2 A}{\cos^2 A}
\][/tex]
### Step 6: Recognize the result
The quotient of [tex]\(\frac{\sin^2 A}{\cos^2 A}\)[/tex] is:
[tex]\[
\frac{\sin^2 A}{\cos^2 A} = \left(\frac{\sin A}{\cos A}\right)^2 = \tan^2 A
\][/tex]
### Final Result:
Thus, the given expression simplifies to:
[tex]\[
\frac{1 + \tan^2 A}{1 + \cot^2 A} = \tan^2 A
\][/tex]
Hence, the simplified form is:
[tex]\[
\boxed{\tan^2 A}
\][/tex]
This is a detailed, step-by-step solution to the problem.
