Trigonometric Expression Simplification
Certainly! The expression given in the image is:
\( \frac{\sin(\frac{3\pi}{2} + \theta) + \cot(-\theta)}{1 - \sin(2\pi - \theta)} \)
Let's simplify the numerator and denominator of this fraction step by step using trigonometric identities:
1. \(\sin(\frac{3\pi}{2} + \theta)\) can be simplified using the identity that \(\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)\):
Since \(\sin(\frac{3\pi}{2}) = -1\) and \(\cos(\theta) = \cos(\theta)\), we get:
\(\sin(\frac{3\pi}{2} + \theta) = \sin(\frac{3\pi}{2})\cos(\theta) + \cos(\frac{3\pi}{2})\sin(\theta) = -1 \cdot \cos(\theta) + 0 \cdot \sin(\theta) = -\cos(\theta)\)
2. \(\cot(-\theta) = \frac{\cos(-\theta)}{\sin(-\theta)}\) can be further simplified using the facts that \(\cos\) is an even function and \(\sin\) is an odd function, which results in \(\cos(-\theta) = \cos(\theta)\) and \(\sin(-\theta) = -\sin(\theta)\):
\(\cot(-\theta) = \frac{\cos(\theta)}{-\sin(\theta)} = -\cot(\theta)\)
Putting these together, the numerator becomes:
\(-\cos(\theta) - \cot(\theta)\)
For the denominator:
1. \(\sin(2\pi - \theta)\) can be simplified using the identity for \(\sin(\pi - x) = \sin(x)\), as \(2\pi - \theta\) is the same as \(\pi - (\pi - \theta)\):
\(\sin(2\pi - \theta) = \sin(\pi - (\pi - \theta)) = \sin(\pi - \theta)\)
Since \(\sin(\pi - x) = \sin(x)\), we get:
\(\sin(2\pi - \theta) = \sin(\theta)\)
So the original expression simplifies to:
\( \frac{-\cos(\theta) - \cot(\theta)}{1 - \sin(\theta)} \)
Now let's simplify further:
Since \(\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\), we can rewrite the numerator in terms of sines and cosines, getting:
\( \frac{-\cos(\theta) - \frac{\cos(\theta)}{\sin(\theta)}}{1 - \sin(\theta)} \)
To combine the terms in the numerator, we get a common denominator of \(\sin(\theta)\):
\( \frac{-\cos(\theta)\sin(\theta) - \cos(\theta)}{\sin(\theta)(1 - \sin(\theta))} \)
Now, we factor out \(-\cos(\theta)\) from the numerator:
\( \frac{-\cos(\theta)(\sin(\theta) + 1)}{\sin(\theta)(1 - \sin(\theta))} \)
Observing that \(\sin(\theta) + 1\) is the additive inverse of \(1 - \sin(\theta)\), we can cancel the corresponding terms in the numerator and denominator, yielding:
\( \frac{-\cos(\theta)}{\sin(\theta)} \)
Finally, this expression can be simplified to
\( -\cot(\theta) \)
So the simplified form of the original expression is \( -\cot(\theta) \).
