Prove tan(θ / 2) = sin θ / (1 + cos θ) for θ in quadrant 1 .

we are asked to prove tan(θ / 2) = sin θ / (1 + cos θ). In this case, tan θ is equal to sin θ / cos θ. we can apply this to the equality. sin θ is equal to square root of (1-cos θ)/2 while cos θ is equal to square root of (1 + cos θ)/2.
Hence, when we replace cos θ with square root of (1-cos θ)/2, we can prove already.

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eudora eudora · Answer 2 · 2019-07-02T21:43:26-04:00

Answer:

Step-by-step explanation:

We have to prove the identity\(tan(\frac{\Theta }{2})=\frac{sin\Theta}{1+cos\Theta }\)

We will take right hand side of the identity

\(\frac{sin\Theta}{1+cos\Theta}=\frac{2sin(\frac{\Theta }{2})cos(\frac{\Theta }{2})}{1+[2cos^{2}(\frac{\Theta }{2})-1]}\)

\(=\frac{2sin(\frac{\Theta }{2})cos(\frac{\Theta }{2})}{2cos^{2}(\frac{\Theta }{2})}\)\(=\frac{sin(\frac{\Theta }{2})}{cos(\frac{\Theta }{2})}\)

\(=tan(\frac{\Theta }{2})\) [ Tan θ will be positive since θ lies in 1st quadrant ]

= L. H. S.

Hence proved.