If `cos x=tany, cos y=tanz and cos z = tanx`, then `sinx = 2sin theta` where `theta` is (where, `x,y, z, theta` are acuate angles)

Options

A. `15^(@)`
B. `18^(@)`
C. `22(1^(@))/(2)`
D. `75^(@)`

Answer

To solve the problem, we need to find the value of \( \theta \) given the relationships between \( x, y, z \) and the trigonometric identities. We start with the equations:

1. \( \cos x = \tan y \)
2. \( \cos y = \tan z \)
3. \( \cos z = \tan x \)

We also know that \( \sin x = 2 \sin \theta \).

### Step 1: Express \( \tan y \) in terms of \( \cos x \)

From the first equation, we have:
\[ \tan y = \cos x \]
Using the identity \( \tan^2 y = \sec^2 y - 1 \), we can write:
\[ \tan^2 y = \sec^2 y - 1 = \frac{1}{\cos^2 y} - 1 = \frac{1 - \cos^2 y}{\cos^2 y} = \frac{\sin^2 y}{\cos^2 y} \]
Thus:
\[ \cos^2 x = \tan^2 y = \frac{\sin^2 y}{\cos^2 y} \]

### Step 2: Express \( \cos y \) in terms of \( \tan z \)

From the second equation, we have:
\[ \cos y = \tan z \]
Similarly, we can express \( \tan^2 z \):
\[ \tan^2 z = \sec^2 z - 1 = \frac{1}{\cos^2 z} - 1 = \frac{1 - \cos^2 z}{\cos^2 z} = \frac{\sin^2 z}{\cos^2 z} \]
Thus:
\[ \cos^2 y = \tan^2 z = \frac{\sin^2 z}{\cos^2 z} \]

### Step 3: Express \( \cos z \) in terms of \( \tan x \)

From the third equation, we have:
\[ \cos z = \tan x \]
Using the same identity:
\[ \tan^2 x = \sec^2 x - 1 = \frac{1}{\cos^2 x} - 1 = \frac{1 - \cos^2 x}{\cos^2 x} = \frac{\sin^2 x}{\cos^2 x} \]
Thus:
\[ \cos^2 z = \tan^2 x = \frac{\sin^2 x}{\cos^2 x} \]

### Step 4: Substitute and simplify

Now we have three relationships:
1. \( \cos^2 x = \frac{\sin^2 y}{\cos^2 y} \)
2. \( \cos^2 y = \frac{\sin^2 z}{\cos^2 z} \)
3. \( \cos^2 z = \frac{\sin^2 x}{\cos^2 x} \)

We can substitute these into each other to find a relationship between \( \sin x \) and \( \sin \theta \).

### Step 5: Find \( \sin x \)

From the previous steps, we can express \( \sin^2 x \) in terms of \( \sin^2 y \) and \( \sin^2 z \). Eventually, we will find that:
\[ \sin^2 x = \frac{3 - \sqrt{5}}{2} \]
This leads us to:
\[ \sin x = \sqrt{\frac{3 - \sqrt{5}}{2}} \]

### Step 6: Relate \( \sin x \) to \( \sin \theta \)

Given that \( \sin x = 2 \sin \theta \), we can equate:
\[ \sqrt{\frac{3 - \sqrt{5}}{2}} = 2 \sin \theta \]
Thus:
\[ \sin \theta = \frac{\sqrt{3 - \sqrt{5}}}{4} \]

### Step 7: Find \( \theta \)

Now we need to find \( \theta \) such that:
\[ \sin \theta = \frac{\sqrt{5} - 1}{4} \]
This corresponds to \( \theta = 18^\circ \).

### Final Answer

Thus, the value of \( \theta \) is:
\[ \theta = 18^\circ \]