Let's solve the given trigonometric expression step by step to simplify it.
Given expression:
[tex]\[ \frac{\sin(180^\circ - y) \cdot \cos(340^\circ - y) \cdot \tan(y)}{\cos(36 \cdot 0 - y) \cdot \cos(180^\circ + y) \cdot \tan(180^\circ + y)} \][/tex]
Step 1: Simplify the angles using trigonometric identities
- [tex]\(\sin(180^\circ - y) = \sin(y)\)[/tex] due to [tex]\(\sin(180^\circ - x) = \sin(x)\)[/tex]
- [tex]\(\cos(340^\circ - y) = \cos(360^\circ - 20^\circ - y) = \cos(-20^\circ - y) = \cos(-y - 20^\circ)\)[/tex] and [tex]\(\cos(-x) = \cos(x)\)[/tex], so [tex]\(\cos(-y - 20^\circ) = \cos(y + 20^\circ)\)[/tex]
- [tex]\(\cos(36 \cdot 0 - y) = \cos(-y) = \cos(y)\)[/tex]
- [tex]\(\cos(180^\circ + y) = -\cos(y)\)[/tex] due to [tex]\(\cos(180^\circ + x) = -\cos(x)\)[/tex]
- [tex]\(\tan(180^\circ + y) = \tan(y)\)[/tex] due to [tex]\(\tan(180^\circ + x) = \tan(x)\)[/tex]
Now, substituting these simplified angles into the expression, we get:
[tex]\[ \frac{\sin(y) \cdot \cos(y + 20^\circ) \cdot \tan(y)}{\cos(y) \cdot (-\cos(y)) \cdot \tan(y)} \][/tex]
Step 2: Simplify the expression further
Separate the numerator and the denominator:
[tex]\[
\text{Numerator: } \sin(y) \cdot \cos(y + 20^\circ) \cdot \tan(y)
\][/tex]
[tex]\[
\text{Denominator: } \cos(y) \cdot (-\cos(y)) \cdot \tan(y)
\][/tex]
Simplifying the denominator:
[tex]\[
\text{Denominator: } -\cos(y)^2 \cdot \tan(y)
\][/tex]
Step 3: Combine everything back
[tex]\[
\frac{\sin(y) \cdot \cos(y + 20^\circ) \cdot \tan(y)}{-\cos(y)^2 \cdot \tan(y)}
\][/tex]
We can cancel [tex]\(\tan(y)\)[/tex] from both the numerator and the denominator:
[tex]\[
\frac{\sin(y) \cdot \cos(y + 20^\circ)}{-\cos(y)^2}
\][/tex]
[tex]\[
\frac{\sin(y) \cdot \cos(y + 20^\circ)}{-\cos(y)^2} = -\frac{\sin(y) \cdot \cos(y + 20^\circ)}{\cos(y)^2}
\][/tex]
This gives us the simplified form of the expression:
[tex]\[
-\sin(y) \cdot \frac{\cos(y + 20^\circ)}{\cos(y)^2}
\][/tex]
For simplicity, you can recognize that [tex]\(20^\circ\)[/tex] is equivalent to [tex]\(\frac{\pi}{9}\)[/tex] radians in trigonometric terms. Thus:
[tex]\[
-\sin(y) \cdot \frac{\cos(y + \frac{\pi}{9})}{\cos(y)^2}
\][/tex]
So, the final simplified expression is:
[tex]\[
-\sin(y) \cdot \cos\left(y + \frac{\pi}{9}\right) \cdot \frac{1}{\cos(y)^2}
\][/tex]
In conclusion, the simplified form of the given trigonometric expression is:
[tex]\[
-\sin(y) \cdot \frac{\cos(y + \frac{\pi}{9})}{\cos(y)^2}
\][/tex]
