Sure, let's establish the identity:
[tex]\[ \frac{\sin(\theta) + \sin(3\theta)}{2 \sin(2\theta)} = \cos(\theta) \][/tex]
To do this, let's follow through with some trigonometric manipulations and simplifications.
1. Express [tex]\(\sin(3\theta)\)[/tex] using trigonometric identities:
We know from trigonometric identities that:
[tex]\[ \sin(3\theta) = 3\sin(\theta) - 4\sin^3(\theta) \][/tex]
2. Express [tex]\(\sin(2\theta)\)[/tex] using trigonometric identities:
Similarly, we know that:
[tex]\[ \sin(2\theta) = 2\sin(\theta)\cos(\theta) \][/tex]
3. Substitute these identities into the original equation:
Substituting [tex]\(\sin(3\theta)\)[/tex] and [tex]\(\sin(2\theta)\)[/tex] into the left-hand side of the given equation:
[tex]\[ \frac{\sin(\theta) + (3\sin(\theta) - 4\sin^3(\theta))}{2 \cdot 2\sin(\theta)\cos(\theta)} \][/tex]
[tex]\[ = \frac{\sin(\theta) + 3\sin(\theta) - 4\sin^3(\theta)}{4\sin(\theta)\cos(\theta)} \][/tex]
4. Combine like terms in the numerator:
[tex]\[ = \frac{4\sin(\theta) - 4\sin^3(\theta)}{4\sin(\theta)\cos(\theta)} \][/tex]
5. Factor out [tex]\(\sin(\theta)\)[/tex] from the numerator:
[tex]\[ = \frac{4\sin(\theta)(1 - \sin^2(\theta))}{4\sin(\theta)\cos(\theta)} \][/tex]
6. Simplify the fraction:
[tex]\[ = \frac{1 - \sin^2(\theta)}{\cos(\theta)} \][/tex]
7. Use the Pythagorean identity:
We know that:
[tex]\[ 1 - \sin^2(\theta) = \cos^2(\theta) \][/tex]
Substituting this into our expression:
[tex]\[ \frac{\cos^2(\theta)}{\cos(\theta)} \][/tex]
8. Simplify the fraction further:
[tex]\[ = \cos(\theta) \][/tex]
So, we have shown through a series of trigonometric manipulations and simplifications that:
[tex]\[ \frac{\sin(\theta) + \sin(3\theta)}{2 \sin(2\theta)} = \cos(\theta) \][/tex]
Thus, the identity is established.
