If x is a positive acute angle and cosx=a, an expression for tanx in terms of a is

Options

A. `(1-a)/(a)`
B. `sqrt(1-a^(2))`
C. `(sqrt(1-a^(2)))/(a)`
D. `(1)/(1-a)`

Answer

To find an expression for \( \tan x \) in terms of \( a \) given that \( \cos x = a \) and \( x \) is a positive acute angle, we can follow these steps:

### Step 1: Use the Pythagorean Identity
We know from trigonometric identities that:
\[ \sin^2 x + \cos^2 x = 1 \]
Since we are given \( \cos x = a \), we can substitute \( a \) into the identity:
\[ \sin^2 x + a^2 = 1 \]

### Step 2: Solve for \( \sin^2 x \)
Rearranging the equation gives us:
\[ \sin^2 x = 1 - a^2 \]

### Step 3: Take the Square Root
To find \( \sin x \), we take the square root of both sides:
\[ \sin x = \sqrt{1 - a^2} \]
Since \( x \) is a positive acute angle, \( \sin x \) is positive, so we only consider the positive root.

### Step 4: Use the Definition of \( \tan x \)
We know that:
\[ \tan x = \frac{\sin x}{\cos x} \]
Substituting the values we have:
\[ \tan x = \frac{\sqrt{1 - a^2}}{a} \]

### Final Expression
Thus, the expression for \( \tan x \) in terms of \( a \) is:
\[ \tan x = \frac{\sqrt{1 - a^2}}{a} \]

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