To determine if the given trigonometric identity [tex]\(\frac{1+\sin \theta - \cos \theta}{1+\sin \theta + \cos \theta} = \tan \frac{\theta}{2}\)[/tex] holds true, we need to verify both sides step by step.
Let's assign:
[tex]\[
\text{Left side} = \frac{1 + \sin \theta - \cos \theta}{1 + \sin \theta + \cos \theta}
\][/tex]
[tex]\[
\text{Right side} = \tan \frac{\theta}{2}
\][/tex]
First, we analyze the left side:
[tex]\[
\frac{1 + \sin \theta - \cos \theta}{1 + \sin \theta + \cos \theta}
\][/tex]
To simplify the left side, let's use the following trigonometric identities and transformations:
1. Sine and Cosine Compound Angles:
Use the identity for compound angles to transform our expression:
[tex]\[
\sin \theta = \sin\left(\theta\right)
\][/tex]
[tex]\[
\cos \theta = \cos\left(\theta\right)
\][/tex]
2. Using the Transformation:
We can express [tex]\(\theta\)[/tex] in terms of [tex]\(\frac{\theta}{2}\)[/tex] which leads us to more convenient forms using double-angle formulas:
[tex]\[
\sin \theta = 2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}
\][/tex]
[tex]\[
\cos \theta = \cos^2 \frac{\theta}{2} - \sin^2 \frac{\theta}{2}
\][/tex]
However, a more effective route to simplify this expression is by using sum-to-product identities and trigonometric properties altogether.
We arrive at the simplified form of the left side:
[tex]\[
\frac{1 + \sin \theta - \cos \theta}{1 + \sin \theta + \cos \theta}
= \frac{(-\sqrt{2}) \cos \left(\theta + \frac{\pi}{4}\right) + 1}{(\sqrt{2}) \sin \left(\theta + \frac{\pi}{4}\right) + 1}
\][/tex]
Now, consider the right side:
[tex]\[
\tan \frac{\theta}{2}
\][/tex]
To check for equivalence, we analyze whether:
[tex]\[
\frac{-\sqrt{2} \cos \left(\theta + \frac{\pi}{4}\right) + 1}{\sqrt{2} \sin \left(\theta + \frac{\pi}{4}\right) + 1} \overset{?}{=} \tan \frac{\theta}{2}
\][/tex]
To determine if the numerator and the denominator simplify in a manner that equals the right side, we observe from our derived forms:
[tex]\[
\frac{-\sqrt{2} \cos \left(\theta + \frac{\pi}{4}\right) + 1}{\sqrt{2} \sin \left(\theta + \frac{\pi}{4}\right) + 1}
\][/tex]
Both simplified forms show that the two expressions do not simplify to exactly the same form. Therefore, upon further evaluation, we realize:
[tex]\[
\frac{1 + \sin \theta - \cos \theta}{1 + \sin \theta + \cos \theta} \neq \tan \frac{\theta}{2}
\][/tex]
Thus, the given trigonometric identity does not hold true and the correct answer is:
[tex]\[
\boxed{\text{False}}
\][/tex]
