If `sinA, cosA and tanA` are in G.P, the `cos^(3)A+cos^(2)A` is equal to :

Options

A. 1
B. 2
C. 4
D. none

Answer

To solve the problem, we need to find the value of \( \cos^3 A + \cos^2 A \) given that \( \sin A, \cos A, \tan A \) are in geometric progression (G.P.).

### Step-by-step Solution:

1. **Understanding G.P. Condition**:
Since \( \sin A, \cos A, \tan A \) are in G.P., we can use the property of G.P. which states that for three terms \( A, B, C \) in G.P., the relationship is given by:
\[ B^2 = A \cdot C \]
Here, we can set \( A = \sin A \), \( B = \cos A \), and \( C = \tan A \).

2. **Applying the G.P. Condition**:
Substituting the terms into the G.P. condition:
\[ \cos^2 A = \sin A \cdot \tan A \]
We know that \( \tan A = \frac{\sin A}{\cos A} \). So, substituting this into the equation gives:
\[ \cos^2 A = \sin A \cdot \left(\frac{\sin A}{\cos A}\right) \]
This simplifies to:
\[ \cos^2 A = \frac{\sin^2 A}{\cos A} \]

3. **Multiplying Both Sides by \( \cos A \)**:
To eliminate the fraction, multiply both sides by \( \cos A \):
\[ \cos^3 A = \sin^2 A \]

4. **Finding \( \cos^3 A + \cos^2 A \)**:
Now we need to find \( \cos^3 A + \cos^2 A \):
\[ \cos^3 A + \cos^2 A = \sin^2 A + \cos^2 A \]

5. **Using the Pythagorean Identity**:
We know from trigonometric identities that:
\[ \sin^2 A + \cos^2 A = 1 \]

6. **Final Result**:
Therefore, we conclude that:
\[ \cos^3 A + \cos^2 A = 1 \]

### Conclusion:
The value of \( \cos^3 A + \cos^2 A \) is \( 1 \).