The equation `tan^(2)phi+tan^(6)phi=tan^(3)phi*sec^(2)phi` is

Options

A. identity for only one value of `phi`
B. not an identity
C. identity for all values of `phi`
D. None of the above

Explanation

To solve the equation \( \tan^2 \phi + \tan^6 \phi = \tan^3 \phi \sec^2 \phi \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \tan^2 \phi + \tan^6 \phi = \tan^3 \phi \sec^2 \phi \]

### Step 2: Use the identity for secant
Recall that \( \sec^2 \phi = 1 + \tan^2 \phi \). We can substitute this into the equation:
\[ \tan^2 \phi + \tan^6 \phi = \tan^3 \phi (1 + \tan^2 \phi) \]

### Step 3: Expand the right-hand side
Expanding the right-hand side gives us:
\[ \tan^2 \phi + \tan^6 \phi = \tan^3 \phi + \tan^5 \phi \]

### Step 4: Rearrange the equation
Now, we rearrange the equation to bring all terms to one side:
\[ \tan^6 \phi - \tan^5 \phi + \tan^2 \phi - \tan^3 \phi = 0 \]

### Step 5: Factor the equation
We can factor out \( \tan^2 \phi \):
\[ \tan^2 \phi (\tan^4 \phi - \tan^3 \phi + 1 - \tan \phi) = 0 \]

### Step 6: Analyze the factors
The equation \( \tan^2 \phi = 0 \) gives us \( \tan \phi = 0 \), which occurs at \( \phi = n\pi \) for any integer \( n \).

The quartic polynomial \( \tan^4 \phi - \tan^3 \phi + 1 - \tan \phi = 0 \) needs to be analyzed further, but for the purpose of this problem, we focus on the first factor.

### Conclusion
Since the equation has solutions for specific values of \( \phi \) (like \( \phi = n\pi \)), it is not an identity for all values of \( \phi \).

### Final Answer
The equation is **not an identity**.

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