Which is equivalent to `(sin^2 alpha + cos^2 alpha )/(cos alpha )` ?

Options

A. `sinalpha`
B. `tan alpha `
C. `sec alpha `
D. `cos alpha `

Answer

To solve the expression \((\sin^2 \alpha + \cos^2 \alpha) / (\cos \alpha)\), we can follow these steps:

### Step 1: Use the Pythagorean Identity
We know from trigonometric identities that:
\[ \sin^2 \alpha + \cos^2 \alpha = 1 \]
So, we can substitute this identity into our expression.

### Step 2: Substitute the Identity
Now, substituting \(1\) for \(\sin^2 \alpha + \cos^2 \alpha\) in the expression gives:
\[ \frac{\sin^2 \alpha + \cos^2 \alpha}{\cos \alpha} = \frac{1}{\cos \alpha} \]

### Step 3: Simplify the Expression
The expression \(\frac{1}{\cos \alpha}\) can be recognized as the secant function:
\[ \frac{1}{\cos \alpha} = \sec \alpha \]

### Final Answer
Thus, the expression \((\sin^2 \alpha + \cos^2 \alpha) / (\cos \alpha)\) is equivalent to:
\[ \sec \alpha \]

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