If `sinA, cosA and tanA` are in G.P., then `cos^(3)A+cos^(2)A=?`

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem where \( \sin A, \cos A, \) and \( \tan A \) are in geometric progression (G.P.), we can follow these steps:

### Step 1: Understanding the condition for G.P.
If three terms \( a, b, c \) are in G.P., then the ratio of the second term to the first term is equal to the ratio of the third term to the second term. Mathematically, this can be expressed as:
\[ \frac{b}{a} = \frac{c}{b} \]
In our case, we have:
- \( a = \sin A \)
- \( b = \cos A \)
- \( c = \tan A \)

### Step 2: Set up the equation
Using the G.P. condition:
\[ \frac{\cos A}{\sin A} = \frac{\tan A}{\cos A} \]
Substituting \( \tan A = \frac{\sin A}{\cos A} \):
\[ \frac{\cos A}{\sin A} = \frac{\frac{\sin A}{\cos A}}{\cos A} \]
This simplifies to:
\[ \frac{\cos A}{\sin A} = \frac{\sin A}{\cos^2 A} \]

### Step 3: Cross-multiply to eliminate the fractions
Cross-multiplying gives:
\[ \cos^2 A \cdot \cos A = \sin^2 A \]
This can be rewritten as:
\[ \cos^3 A = \sin^2 A \]

### Step 4: Use the Pythagorean identity
We know from the Pythagorean identity that:
\[ \sin^2 A + \cos^2 A = 1 \]
Substituting \( \sin^2 A \) with \( \cos^3 A \):
\[ \cos^3 A + \cos^2 A = 1 \]

### Step 5: Conclusion
Thus, we find that:
\[ \cos^3 A + \cos^2 A = 1 \]

### Final Answer
The value of \( \cos^3 A + \cos^2 A \) is \( \boxed{1} \).