`(1+tan^(2)A)+(1+1/(tan^(2)A))`is equal to

Options

A. `1/(sin^(2)A-sin^(4)A)`
B. `1/(sin^(2)A+sin^(4)A)`
C. `(cos^(2)A)/(sinA+sin^(2)A)`
D. `(cos^(2)A)/(sinA-sin^(2)A)`

Answer

To solve the expression \((1 + \tan^2 A) + \left(1 + \frac{1}{\tan^2 A}\right)\), we will simplify it step by step.

### Step 1: Rewrite the expression
We start with the given expression:
\[ (1 + \tan^2 A) + \left(1 + \frac{1}{\tan^2 A}\right) \]

### Step 2: Combine the terms
We can combine the constants:
\[ 1 + \tan^2 A + 1 + \frac{1}{\tan^2 A} = 2 + \tan^2 A + \frac{1}{\tan^2 A} \]

### Step 3: Use the identity for \(\tan^2 A\)
We know from trigonometric identities that:
\[ 1 + \tan^2 A = \sec^2 A \]
Thus, we can write:
\[ \tan^2 A + \frac{1}{\tan^2 A} \]

### Step 4: Find a common denominator
To combine \(\tan^2 A + \frac{1}{\tan^2 A}\), we find a common denominator:
\[ \tan^2 A + \frac{1}{\tan^2 A} = \frac{\tan^4 A + 1}{\tan^2 A} \]

### Step 5: Substitute back into the expression
Now we substitute this back into our expression:
\[ 2 + \frac{\tan^4 A + 1}{\tan^2 A} \]

### Step 6: Combine the terms
We can rewrite it as:
\[ \frac{2\tan^2 A + \tan^4 A + 1}{\tan^2 A} \]

### Step 7: Factor the numerator
The numerator \(2\tan^2 A + \tan^4 A + 1\) can be factored:
\[ \tan^4 A + 2\tan^2 A + 1 = (\tan^2 A + 1)^2 \]

### Step 8: Final expression
Thus, we have:
\[ \frac{(\tan^2 A + 1)^2}{\tan^2 A} \]

### Step 9: Substitute back the identity
Now substituting back \(\tan^2 A + 1 = \sec^2 A\):
\[ \frac{\sec^4 A}{\tan^2 A} \]

### Step 10: Final simplification
Using the identity \(\tan^2 A = \frac{\sin^2 A}{\cos^2 A}\) and \(\sec^2 A = \frac{1}{\cos^2 A}\):
\[ \frac{1}{\sin^2 A} \]

### Final Answer
Thus, the final answer is:
\[ \frac{1}{\sin^2 A} - \sin^4 A \]