If `x=acosthetacostheta, y=acosthetasinthetaandz=asintheta`, then the value of `(x^(2)+y^(2)+z^(2))` is

Options

A. `2a^(2)`
B. `4a^(2)`
C. `9a^(2)`
D. `a^(2)`

Answer

To solve the problem, we need to find the value of \( x^2 + y^2 + z^2 \) given the equations:

\[ x = a \cos^2 \theta, \quad y = a \cos \theta \sin \theta, \quad z = a \sin \theta \]

### Step 1: Calculate \( x^2 \)

Starting with \( x \):

\[ x = a \cos^2 \theta \]

Now, squaring \( x \):

\[ x^2 = (a \cos^2 \theta)^2 = a^2 \cos^4 \theta \]

### Step 2: Calculate \( y^2 \)

Next, for \( y \):

\[ y = a \cos \theta \sin \theta \]

Now, squaring \( y \):

\[ y^2 = (a \cos \theta \sin \theta)^2 = a^2 \cos^2 \theta \sin^2 \theta \]

### Step 3: Calculate \( z^2 \)

Now, for \( z \):

\[ z = a \sin \theta \]

Now, squaring \( z \):

\[ z^2 = (a \sin \theta)^2 = a^2 \sin^2 \theta \]

### Step 4: Combine \( x^2, y^2, z^2 \)

Now, we will add \( x^2, y^2, z^2 \):

\[ x^2 + y^2 + z^2 = a^2 \cos^4 \theta + a^2 \cos^2 \theta \sin^2 \theta + a^2 \sin^2 \theta \]

### Step 5: Factor out \( a^2 \)

Factoring out \( a^2 \):

\[ x^2 + y^2 + z^2 = a^2 \left( \cos^4 \theta + \cos^2 \theta \sin^2 \theta + \sin^2 \theta \right) \]

### Step 6: Simplify the expression

Now, we can simplify the expression inside the parentheses. Notice that:

\[ \cos^4 \theta + \sin^2 \theta = \cos^4 \theta + \sin^2 \theta + \cos^2 \theta \sin^2 \theta \]

This can be rewritten as:

\[ \cos^4 \theta + \sin^2 \theta = \cos^4 \theta + \sin^2 \theta + \cos^2 \theta \sin^2 \theta = \cos^2 \theta (\cos^2 \theta + \sin^2 \theta) + \sin^2 \theta \]

Using the identity \( \cos^2 \theta + \sin^2 \theta = 1 \):

\[ = \cos^2 \theta \cdot 1 + \sin^2 \theta = \cos^2 \theta + \sin^2 \theta = 1 \]

So, we have:

\[ x^2 + y^2 + z^2 = a^2 \cdot 1 = a^2 \]

### Final Answer

Thus, the value of \( x^2 + y^2 + z^2 \) is:

\[ \boxed{a^2} \]