`theta_(1),theta_(2),theta_(3)` are angles of `1^(st)` quadrant if `tan theta_(1) = cos theta_(1), tan theta_(2) = cosec theta_(2), cos theta_(3)=theta_(3)`. Which of the following is not true ?

Options

A. `theta_(1)lt theta_(2)`
B. `theta_(1)lt theta_(3)`
C. `theta_(3)lt theta_(1)`
D. `theta_(3)lt theta_(2)`

Explanation

To solve the problem, we need to analyze the given conditions for the angles \( \theta_1, \theta_2, \) and \( \theta_3 \) in the first quadrant.

### Step 1: Analyze \( \theta_1 \)

Given:
\[ \tan \theta_1 = \cos \theta_1 \]

We know that:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]

Thus, we can rewrite the equation:
\[ \frac{\sin \theta_1}{\cos \theta_1} = \cos \theta_1 \]

Multiplying both sides by \( \cos \theta_1 \) (assuming \( \cos \theta_1 \neq 0 \)):
\[ \sin \theta_1 = \cos^2 \theta_1 \]

Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we can express \( \sin \theta_1 \) in terms of \( \cos \theta_1 \):
\[ \sin^2 \theta_1 = 1 - \cos^2 \theta_1 \]

Substituting \( \sin \theta_1 = \cos^2 \theta_1 \):
\[ (\cos^2 \theta_1)^2 = 1 - \cos^2 \theta_1 \]
\[ \cos^4 \theta_1 + \cos^2 \theta_1 - 1 = 0 \]

Let \( x = \cos^2 \theta_1 \):
\[ x^2 + x - 1 = 0 \]

Using the quadratic formula:
\[ x = \frac{-1 \pm \sqrt{1 + 4}}{2} = \frac{-1 \pm \sqrt{5}}{2} \]

Since \( x = \cos^2 \theta_1 \) must be non-negative, we take:
\[ x = \frac{-1 + \sqrt{5}}{2} \]

### Step 2: Analyze \( \theta_2 \)

Given:
\[ \tan \theta_2 = \csc \theta_2 \]

We can rewrite this as:
\[ \tan \theta_2 = \frac{1}{\sin \theta_2} \]

Thus:
\[ \frac{\sin \theta_2}{\cos \theta_2} = \frac{1}{\sin \theta_2} \]

Cross-multiplying gives:
\[ \sin^2 \theta_2 = \cos \theta_2 \]

Using \( \sin^2 \theta_2 + \cos^2 \theta_2 = 1 \):
\[ 1 - \cos^2 \theta_2 = \cos \theta_2 \]
\[ \cos^2 \theta_2 + \cos \theta_2 - 1 = 0 \]

Let \( y = \cos \theta_2 \):
\[ y^2 + y - 1 = 0 \]

Using the quadratic formula:
\[ y = \frac{-1 \pm \sqrt{1 + 4}}{2} = \frac{-1 \pm \sqrt{5}}{2} \]

Taking the positive root:
\[ y = \frac{-1 + \sqrt{5}}{2} \]

### Step 3: Analyze \( \theta_3 \)

Given:
\[ \cos \theta_3 = \theta_3 \]

The function \( y = \cos x \) intersects \( y = x \) only at one point in the first quadrant, which is approximately \( \theta_3 \approx 0.739 \) (the fixed point of the cosine function).

### Step 4: Compare the Angles

From the analysis:
- \( \theta_1 < \theta_3 \) since \( \cos^2 \theta_1 \) is less than 1.
- \( \theta_2 > 45^\circ \) since \( \tan \theta_2 > 1 \).

### Conclusion

The relationships we have are:
- \( \theta_1 < \theta_3 \)
- \( \theta_2 > 45^\circ \)

Thus, the statement that is **not true** is:
\[ \theta_3 < \theta_1 \]

### Final Answer
The option that is not true is \( \theta_3 < \theta_1 \).