`cos 2 theta ` is not equal to

Options

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

Answer

To solve the problem regarding the expression for \( \cos 2\theta \) and to determine which value it is not equal to, we can follow these steps:

### Step 1: Recall the formula for \( \cos 2\theta \)
The standard formula for \( \cos 2\theta \) is:
\[ \cos 2\theta = \cos^2\theta - \sin^2\theta \]

### Step 2: Use the Pythagorean identity
We know from the Pythagorean identity that:
\[ \sin^2\theta + \cos^2\theta = 1 \]
From this, we can express \( \sin^2\theta \) in terms of \( \cos^2\theta \):
\[ \sin^2\theta = 1 - \cos^2\theta \]

### Step 3: Substitute \( \sin^2\theta \) into the \( \cos 2\theta \) formula
Substituting \( \sin^2\theta \) into the \( \cos 2\theta \) formula gives:
\[ \cos 2\theta = \cos^2\theta - (1 - \cos^2\theta) \]
This simplifies to:
\[ \cos 2\theta = \cos^2\theta - 1 + \cos^2\theta = 2\cos^2\theta - 1 \]

### Step 4: Consider the alternative form of \( \cos 2\theta \)
There is another form of \( \cos 2\theta \) which is:
\[ \cos 2\theta = 1 - 2\sin^2\theta \]
This can also be derived from the Pythagorean identity.

### Step 5: Analyze the options
Now, we have two forms of \( \cos 2\theta \):
1. \( \cos 2\theta = 2\cos^2\theta - 1 \)
2. \( \cos 2\theta = 1 - 2\sin^2\theta \)

Given the options in the question, we need to determine which expression \( \cos 2\theta \) is not equal to.

### Step 6: Evaluate the options
Let's say the options are:
A) \( 2\cos^2\theta - 1 \)
B) \( 1 - 2\sin^2\theta \)
C) \( \sin^2\theta \)
D) \( 1 \)

Since \( \cos 2\theta \) can take values based on the angle \( \theta \), we can conclude that \( \cos 2\theta \) is not equal to \( \sin^2\theta \) or \( 1 \) in general, depending on the value of \( \theta \).

### Conclusion
Thus, the value that \( \cos 2\theta \) is not equal to is most likely option C or D, depending on the context of the question.