Question - Trigonometric Expression Simplification
mathtrigonometric identitiestrigonometric expression simplificationsimplify trigonometric expressionstep by step simplification
Solution:
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 $$ -\cot(\theta) $$.
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