Prove trigonometric an identity \frac{2 \cos \alpha-1}{\sqrt 3 -2 \sin \alpha}=\tan(\frac{\alpha}{2}+\frac{\pi}{6})

Daniell Phillips 2022-01-17 Answered
Prove trigonometric an identity
2cosα132sinα=tan(α2+π6)
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Expert Answer

Joseph Fair
Answered 2022-01-18 Author has 34 answers
Using Prosthaphaeresis Formulas
12=sinπ6=cosπ3
32=cosπ6=sinπ3
cosαcosπ3=2sin(π6α2)sin(π6+α2)
sinπ3sinα=2sin(π6α2)cos(π6+α2)
Remember we need sin(π6α2)0 to reach at the identity required
Alternatively, we can use Weierstrass Substitution in the left hand side and use tan(A+B) formula
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RizerMix
Answered 2022-01-19 Author has 438 answers

1(tanx2+1)2cos2x2=1(tan2x2+2tanx2+1)cos2x2=1sin2x2+2sinx2cosx2+cos2x2=11+2sinx2cosx2=11+sinx

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