If`theta` is eliminated from the equations `costheta-sintheta=b` and `cos3theta+sin3theta=a`,find the eliminant

Answer

To eliminate θ from the equations \( \cos \theta - \sin \theta = b \) and \( \cos 3\theta + \sin 3\theta = a \), we can follow these steps:

### Step 1: Square the first equation
Starting with the first equation:
\[ \cos \theta - \sin \theta = b \]
Square both sides:
\[ (\cos \theta - \sin \theta)^2 = b^2 \]
Expanding the left side:
\[ \cos^2 \theta - 2\cos \theta \sin \theta + \sin^2 \theta = b^2 \]
Using the identity \( \cos^2 \theta + \sin^2 \theta = 1 \):
\[ 1 - 2\cos \theta \sin \theta = b^2 \]
Rearranging gives:
\[ 2\cos \theta \sin \theta = 1 - b^2 \]
Thus, we find:
\[ \cos \theta \sin \theta = \frac{1 - b^2}{2} \]

### Step 2: Use the second equation
Now, consider the second equation:
\[ \cos 3\theta + \sin 3\theta = a \]
Using the formulas for \( \cos 3\theta \) and \( \sin 3\theta \):
\[ \cos 3\theta = 4\cos^3 \theta - 3\cos \theta \]
\[ \sin 3\theta = 3\sin \theta - 4\sin^3 \theta \]
Thus, we can write:
\[ (4\cos^3 \theta - 3\cos \theta) + (3\sin \theta - 4\sin^3 \theta) = a \]

### Step 3: Factor out \( \cos \theta - \sin \theta \)
We can factor this expression:
\[ 4\cos^3 \theta - 4\sin^3 \theta - 3(\cos \theta - \sin \theta) = a \]
Using the identity for the difference of cubes:
\[ a = (\cos \theta - \sin \theta)(4(\cos^2 \theta + \cos \theta \sin \theta + \sin^2 \theta) - 3) \]
Since \( \cos^2 \theta + \sin^2 \theta = 1 \):
\[ a = (\cos \theta - \sin \theta)(4(1 + \cos \theta \sin \theta) - 3) \]
Substituting \( \cos \theta \sin \theta = \frac{1 - b^2}{2} \):
\[ a = (\cos \theta - \sin \theta)(4(1 + \frac{1 - b^2}{2}) - 3) \]
This simplifies to:
\[ a = (\cos \theta - \sin \theta)(4 + 2 - 2b^2 - 3) \]
Thus:
\[ a = (\cos \theta - \sin \theta)(3 - 2b^2) \]

### Step 4: Substitute \( \cos \theta - \sin \theta \)
Now substituting \( \cos \theta - \sin \theta = b \):
\[ a = b(3 - 2b^2) \]
Rearranging gives:
\[ 2b^3 - 3b + a = 0 \]

### Final Result
The eliminant is:
\[ 2b^3 - 3b + a = 0 \]