If `sin theta = x and sec theta = y`, then `tantheta` is:

Options

A. xy
B. `(x)/(y)`
C. `(y)/(x)`
D. `(1)/(xy)`

Answer

To find the value of \( \tan \theta \) given that \( \sin \theta = x \) and \( \sec \theta = y \), we can follow these steps:

### Step 1: Understand the definitions
We know that:
- \( \sin \theta = x \)
- \( \sec \theta = y \)

The secant function is defined as the reciprocal of the cosine function:
\[ \sec \theta = \frac{1}{\cos \theta} \]
Thus, we can express \( \cos \theta \) in terms of \( y \):
\[ \cos \theta = \frac{1}{y} \]

### Step 2: Write the formula for \( \tan \theta \)
The tangent function is defined as the ratio of sine to cosine:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]

### Step 3: Substitute the known values
Now, substitute the values of \( \sin \theta \) and \( \cos \theta \):
\[ \tan \theta = \frac{x}{\frac{1}{y}} \]

### Step 4: Simplify the expression
When dividing by a fraction, we multiply by its reciprocal:
\[ \tan \theta = x \cdot y \]

### Conclusion
Thus, the value of \( \tan \theta \) is:
\[ \tan \theta = xy \]

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