Proving trigonometric idencities \( \frac{1+\cos \theta}{\sin \theta}+\frac{\sin \theta}{1+\cos \theta}=\frac{2}{\sin \theta} \)
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To prove the identity, start by simplifying the left-hand side: \[ \frac{1+\cos \theta}{\sin \theta} + \frac{\sin \theta}{1+\cos \theta} = \frac{(1+\cos \theta)^2 + \sin^2 \theta}{\sin \theta(1+\cos \theta)} \] Next, expand the numerator: \[ (1+\cos \theta)^2 = 1 + 2\cos \theta + \cos^2 \theta \] \[ \sin^2 \theta = 1 - \cos^2 \theta \] Combining these gives: \[ 1 + 2\cos \theta + \cos^2 \theta + 1 - \cos^2 \theta = 2 + 2\cos \theta \] Thus, the left-hand side becomes: \[ \frac{2(1+\cos \theta)}{\sin \theta(1+\cos \theta)} = \frac{2}{\sin \theta} \] This matches the right-hand side, confirming the identity: \[ \frac{1+\cos \theta}{\sin \theta} + \frac{\sin \theta}{1+\cos \theta} = \frac{2}{\sin \theta} \] To make the process of proving trigonometric identities more engaging, remember that persistence pays off! Don’t shy away from rewriting terms in different forms or using identities like Pythagorean or reciprocal ones. Another fun tip: always keep a well-stocked arsenal of common trigonometric identities in your toolbox. They’re your trusty sidekicks! Once you become comfortable with them, the process becomes as smooth as riding a bike downhill!
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