If `(a^(2)+1)/(2a)=costheta, then (a^(6)+1)/(2a^(3))=`

Options

A. `cos^(2) theta`
B. `cos^(3)theta`
C. `cos 2 theta`
D. `cos 3theta`

Answer

To solve the problem, we start with the given equation:

\[ \frac{a^2 + 1}{2a} = \cos \theta \]

We need to find the value of:

\[ \frac{a^6 + 1}{2a^3} \]

### Step 1: Rewrite the given equation

From the given equation, we can manipulate it:

\[ \frac{a^2 + 1}{2a} = \cos \theta \]

Multiplying both sides by \(2a\):

\[ a^2 + 1 = 2a \cos \theta \]

### Step 2: Rearranging the equation

Now, we can rearrange this equation to isolate \(a^2\):

\[ a^2 = 2a \cos \theta - 1 \]

### Step 3: Finding \(a + \frac{1}{a}\)

Next, we want to find \(a + \frac{1}{a}\). We can express this in terms of \(\cos \theta\):

\[ \frac{a^2 + 1}{a} = 2 \cos \theta \]

Thus,

\[ a + \frac{1}{a} = 2 \cos \theta \]

### Step 4: Cubing both sides

Now, we will cube \(a + \frac{1}{a}\):

\[ \left(a + \frac{1}{a}\right)^3 = (2 \cos \theta)^3 \]

Using the identity \((x + y)^3 = x^3 + y^3 + 3xy(x + y)\):

\[ a^3 + \frac{1}{a^3} + 3 \cdot a \cdot \frac{1}{a} \cdot (a + \frac{1}{a}) = 8 \cos^3 \theta \]

This simplifies to:

\[ a^3 + \frac{1}{a^3} + 3(2 \cos \theta) = 8 \cos^3 \theta \]

### Step 5: Isolating \(a^3 + \frac{1}{a^3}\)

Now, we can isolate \(a^3 + \frac{1}{a^3}\):

\[ a^3 + \frac{1}{a^3} = 8 \cos^3 \theta - 6 \cos \theta \]

### Step 6: Finding \(a^6 + 1\)

Now we can express \(a^6 + 1\) in terms of \(a^3\):

\[ a^6 + 1 = (a^3)^2 + 1 = (a^3 + \frac{1}{a^3})^2 - 2 \]

Substituting \(a^3 + \frac{1}{a^3}\):

\[ a^6 + 1 = \left(8 \cos^3 \theta - 6 \cos \theta\right)^2 - 2 \]

### Step 7: Finding \(\frac{a^6 + 1}{2a^3}\)

Finally, we need to find:

\[ \frac{a^6 + 1}{2a^3} = \frac{(8 \cos^3 \theta - 6 \cos \theta)^2 - 2}{2a^3} \]

Using the identity \(a^3 = \frac{(8 \cos^3 \theta - 6 \cos \theta)}{2}\):

\[ \frac{a^6 + 1}{2a^3} = 4 \cos^3 \theta - 3 \cos \theta \]

### Step 8: Recognizing the identity

The expression \(4 \cos^3 \theta - 3 \cos \theta\) is the identity for \(\cos(3\theta)\):

\[ \frac{a^6 + 1}{2a^3} = \cos(3\theta) \]

Thus, the final answer is:

\[ \frac{a^6 + 1}{2a^3} = \cos(3\theta) \]