prove tan (theta/2)= (sin(theta))/(1+cos(theta))for theta in quadrant 1 by providing the calculation along with the reason.
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To prove the identity tan(theta/2) = sin(theta) / (1 + cos(theta)) for theta in quadrant 1, we will use the half-angle identities and the Pythagorean identity.
1. Start with the half-angle identity for tangent: tan(theta/2) = (1 - cos(theta)) / sin(theta).
2. Since theta is in quadrant 1, both sin(theta) and cos(theta) are positive.
3. Divide both the numerator and denominator of the identity in step 1 by (1 + cos(theta)):
tan(theta/2) = [(1 - cos(theta)) / sin(theta)] * [(1 + cos(theta)) / (1 + cos(theta))]
4. Simplify the numerator:
(1 - cos(theta)) * (1 + cos(theta)) = 1 - cos^2(theta) = sin^2(theta) (using the Pythagorean identity sin^2(theta) + cos^2(theta) = 1)
Therefore, the numerator simplifies to sin^2(theta).
5. Substitute the simplified numerator and rearrange the terms:
tan(theta/2) = [sin^2(theta)] / [sin(theta) * (1 + cos(theta))]
6. Cancel out sin(theta) from the numerator and denominator:
tan(theta/2) = sin(theta) / (1 + cos(theta))
Hence, tan(theta/2) = sin(theta) / (1 + cos(theta)) for theta in quadrant 1.