If `cos^2 theta- sin^2 theta = tan^2 phi` then which of the following is true ?

Options

A. `cos theta cos phi = 1`
B. `cos^2 phi - sin^2 phi = tan^2 theta`
C. `cos^2 phi - sin^2 phi = cot^2 theta`
D. `cos theta cos phi = sqrt(2)`

Answer

To solve the equation \( \cos^2 \theta - \sin^2 \theta = \tan^2 \phi \) and determine which of the given options is true, we can follow these steps:

### Step 1: Rewrite the given equation
The equation provided is:
\[ \cos^2 \theta - \sin^2 \theta = \tan^2 \phi \]

### Step 2: Use the identity for tangent
Recall that:
\[ \tan^2 \phi = \frac{\sin^2 \phi}{\cos^2 \phi} \]
Thus, we can rewrite the equation as:
\[ \cos^2 \theta - \sin^2 \theta = \frac{\sin^2 \phi}{\cos^2 \phi} \]

### Step 3: Express \(\cos^2 \theta\) in terms of \(\sin^2 \phi\)
We can rearrange the equation:
\[ \cos^2 \theta = \sin^2 \phi + \sin^2 \theta \]
Using the identity \(\sin^2 \theta = 1 - \cos^2 \theta\), we can substitute:
\[ \cos^2 \theta = \sin^2 \phi + (1 - \cos^2 \theta) \]
This simplifies to:
\[ 2 \cos^2 \theta = \sin^2 \phi + 1 \]
Thus,
\[ \cos^2 \theta = \frac{\sin^2 \phi + 1}{2} \]

### Step 4: Use the Pythagorean identity
Using the identity \( \sin^2 \phi + \cos^2 \phi = 1 \), we can express \(\sin^2 \phi\) in terms of \(\cos^2 \phi\):
\[ \sin^2 \phi = 1 - \cos^2 \phi \]
Substituting this into our equation gives:
\[ \cos^2 \theta = \frac{(1 - \cos^2 \phi) + 1}{2} \]
This simplifies to:
\[ \cos^2 \theta = \frac{2 - \cos^2 \phi}{2} \]

### Step 5: Analyze the options
Now we can analyze the options given in the question. We need to check which of the following statements is true based on our derived equations.

1. \( \cos \theta \cdot \cos \phi = 1 \)
2. \( \cos^2 \phi - \sin^2 \phi = \tan^2 \theta \)
3. \( \cos^2 \phi - \sin^2 \phi = \cot^2 \theta \)
4. \( \cos \theta \cdot \cos \phi = \sqrt{2} \)

### Conclusion
From the derived equations, we can check which statement holds true.

After evaluating the options, we find that:
- The correct option is \( \cos^2 \phi - \sin^2 \phi = \tan^2 \theta \).