3. Prove that a. $$ \frac{1+\cos heta}{\sin heta}+\frac{\sin heta}{1+\cos heta}=\frac{2}{\sin heta} $$

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3. Prove that a. $$ \frac{1+\cos 	heta}{\sin 	heta}+\frac{\sin 	heta}{1+\cos 	heta}=\frac{2}{\sin 	heta} $$

UpStudy AI · Accepted Answer

1. Express the left-hand side as a single fraction:
   $$
   \frac{1+\cos \theta}{\sin \theta}+\frac{\sin \theta}{1+\cos \theta} = \frac{(1+\cos \theta)^2+\sin^2 \theta}{\sin \theta (1+\cos \theta)}
   $$

2. Expand $$(1+\cos \theta)^2$$:
   $$
   (1+\cos \theta)^2 = 1+2\cos \theta+\cos^2 \theta
   $$

3. Substitute the expansion into the numerator:
   $$
   (1+\cos \theta)^2+\sin^2 \theta = 1+2\cos \theta+\cos^2 \theta+\sin^2 \theta
   $$

4. Use the Pythagorean identity $$\cos^2 \theta + \sin^2 \theta = 1$$:
   $$
   1+2\cos \theta+\cos^2 \theta+\sin^2 \theta = 1+2\cos \theta+1 = 2+2\cos \theta = 2(1+\cos \theta)
   $$

5. Replace the numerator in the fraction:
   $$
   \frac{(1+\cos \theta)^2+\sin^2 \theta}{\sin \theta (1+\cos \theta)} = \frac{2(1+\cos \theta)}{\sin \theta (1+\cos \theta)}
   $$

6. Cancel the common factor $$(1+\cos \theta)$$ (provided $$\cos \theta \neq -1$$, so that the denominator is nonzero):
   $$
   \frac{2(1+\cos \theta)}{\sin \theta (1+\cos \theta)} = \frac{2}{\sin \theta}
   $$

Thus, we have proved that
$$
\frac{1+\cos \theta}{\sin \theta}+\frac{\sin \theta}{1+\cos \theta}=\frac{2}{\sin \theta}.
$$
$$ \frac{1+\cos \theta}{\sin \theta}+\frac{\sin \theta}{1+\cos \theta}=\frac{2}{\sin \theta} $$

3. Prove that a. \( \frac{1+\cos \theta}{\sin \theta}+\frac{\sin \theta}{1+\cos \theta}=\frac{2}{\sin \theta} \)

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