If `cos9alpha=sinalpha` and `9alpha lt 90^(@)`, then the value of `tan5alpha` is

Options

A. `1/sqrt(3)`
B. `sqrt(3)`
C. `1`
D. `0`

Explanation

To solve the problem where \( \cos(9\alpha) = \sin(\alpha) \) and \( 9\alpha < 90^\circ \), and we need to find the value of \( \tan(5\alpha) \), we can follow these steps:

### Step 1: Use the identity for sine and cosine
We know that \( \sin(\theta) = \cos(90^\circ - \theta) \). Therefore, we can rewrite the equation:
\[ \cos(9\alpha) = \sin(\alpha) \implies \cos(9\alpha) = \cos(90^\circ - \alpha) \]

### Step 2: Set the angles equal
Since the cosines are equal, we can set the angles equal to each other:
\[ 9\alpha = 90^\circ - \alpha \]

### Step 3: Solve for \( \alpha \)
Now, we can solve for \( \alpha \):
\[ 9\alpha + \alpha = 90^\circ \]
\[ 10\alpha = 90^\circ \]
\[ \alpha = \frac{90^\circ}{10} = 9^\circ \]

### Step 4: Find \( 5\alpha \)
Now that we have \( \alpha \), we can find \( 5\alpha \):
\[ 5\alpha = 5 \times 9^\circ = 45^\circ \]

### Step 5: Calculate \( \tan(5\alpha) \)
Finally, we can find \( \tan(5\alpha) \):
\[ \tan(5\alpha) = \tan(45^\circ) = 1 \]

Thus, the value of \( \tan(5\alpha) \) is \( 1 \).

### Final Answer:
\[ \tan(5\alpha) = 1 \]

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