`cos 20 theta ` in terms of ` sin 5 theta .`

Answer

To express \( \cos 20\theta \) in terms of \( \sin 5\theta \), we can follow these steps:

### Step 1: Use the double angle identity for cosine
We start with the identity:
\[ \cos 2\theta = 2\cos^2 \theta - 1 \]
We can express \( \cos 20\theta \) as:
\[ \cos 20\theta = \cos(4 \cdot 5\theta) = \cos(2 \cdot 10\theta) = 2\cos^2(10\theta) - 1 \]

### Step 2: Express \( \cos 10\theta \) using the double angle identity again
Now we need to express \( \cos 10\theta \):
\[ \cos 10\theta = \cos(2 \cdot 5\theta) = 2\cos^2(5\theta) - 1 \]
Substituting this back into our expression for \( \cos 20\theta \):
\[ \cos 20\theta = 2(2\cos^2(5\theta) - 1)^2 - 1 \]

### Step 3: Expand the expression
Now we expand \( (2\cos^2(5\theta) - 1)^2 \):
\[ (2\cos^2(5\theta) - 1)^2 = 4\cos^4(5\theta) - 4\cos^2(5\theta) + 1 \]
Substituting this back into the equation for \( \cos 20\theta \):
\[ \cos 20\theta = 2(4\cos^4(5\theta) - 4\cos^2(5\theta) + 1) - 1 \]
\[ = 8\cos^4(5\theta) - 8\cos^2(5\theta) + 2 - 1 \]
\[ = 8\cos^4(5\theta) - 8\cos^2(5\theta) + 1 \]

### Step 4: Use the identity \( \cos^2(5\theta) = 1 - \sin^2(5\theta) \)
Now, we can express \( \cos^2(5\theta) \) in terms of \( \sin(5\theta) \):
\[ \cos^2(5\theta) = 1 - \sin^2(5\theta) \]
Substituting this into our expression:
\[ \cos 20\theta = 8(1 - \sin^2(5\theta))^2 - 8(1 - \sin^2(5\theta)) + 1 \]

### Step 5: Expand and simplify
Now we expand \( (1 - \sin^2(5\theta))^2 \):
\[ (1 - \sin^2(5\theta))^2 = 1 - 2\sin^2(5\theta) + \sin^4(5\theta) \]
Substituting this back:
\[ \cos 20\theta = 8(1 - 2\sin^2(5\theta) + \sin^4(5\theta)) - 8(1 - \sin^2(5\theta)) + 1 \]
\[ = 8 - 16\sin^2(5\theta) + 8\sin^4(5\theta) - 8 + 8\sin^2(5\theta) + 1 \]
\[ = 8\sin^4(5\theta) - 8\sin^2(5\theta) + 1 \]

### Final Expression
Thus, we have expressed \( \cos 20\theta \) in terms of \( \sin 5\theta \):
\[ \cos 20\theta = 8\sin^4(5\theta) - 8\sin^2(5\theta) + 1 \]