Solve : `(1- tan^2 theta)/(tan^2 theta)`

Options

A. `1- cot^2 theta`
B. 2
C. 0
D. `cot^2 theta -1`

Answer

To solve the expression \((1 - \tan^2 \theta) / \tan^2 \theta\), we can follow these steps:

### Step 1: Rewrite \(\tan^2 \theta\)
We know that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), so \(\tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta}\).

### Step 2: Substitute \(\tan^2 \theta\) in the expression
Substituting \(\tan^2 \theta\) into the expression gives us:
\[ \frac{1 - \tan^2 \theta}{\tan^2 \theta} = \frac{1 - \frac{\sin^2 \theta}{\cos^2 \theta}}{\frac{\sin^2 \theta}{\cos^2 \theta}} \]

### Step 3: Simplify the numerator
To simplify the numerator \(1 - \frac{\sin^2 \theta}{\cos^2 \theta}\), we can write \(1\) as \(\frac{\cos^2 \theta}{\cos^2 \theta}\):
\[ 1 - \frac{\sin^2 \theta}{\cos^2 \theta} = \frac{\cos^2 \theta - \sin^2 \theta}{\cos^2 \theta} \]

### Step 4: Substitute back into the expression
Now substituting back into the expression, we have:
\[ \frac{\frac{\cos^2 \theta - \sin^2 \theta}{\cos^2 \theta}}{\frac{\sin^2 \theta}{\cos^2 \theta}} = \frac{\cos^2 \theta - \sin^2 \theta}{\sin^2 \theta} \]

### Step 5: Separate the fractions
We can separate the fractions:
\[ \frac{\cos^2 \theta}{\sin^2 \theta} - \frac{\sin^2 \theta}{\sin^2 \theta} \]

### Step 6: Simplify the fractions
This simplifies to:
\[ \cot^2 \theta - 1 \]

### Final Answer
Thus, the final result of the expression \((1 - \tan^2 \theta) / \tan^2 \theta\) is:
\[ \cot^2 \theta - 1 \]

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