How i prove this ?

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How i prove this ?

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Answer 1 · 2018-04-16T09:14:24

Please see below.

Explanation:

As $sin (α \pm β) = sin α cos β \pm cos α sin β$ $sin(alpha+-beta)=sinalphacosbeta+-cosalphasinbeta$

and $cos (α \pm β) = cos α cos β \mp sin α sin β$ $cos(alpha+-beta)=cosalphacosbeta∓sinalphasinbeta$

Hence $\frac{sin (α + β) - 2 sin α cos β}{2 sin α sin β + cos (α + β)}$ $(sin(alpha+beta)-2sinalphacosbeta)/(2sinalphasinbeta+cos(alpha+beta))$

= $\frac{sin α cos β + cos α sin β - 2 sin α cos β}{2 sin α sin β + cos α cos β - sin α sin β}$ $(sinalphacosbeta+cosalphasinbeta-2sinalphacosbeta)/(2sinalphasinbeta+cosalphacosbeta-sinalphasinbeta)$

= $\frac{sin β cos α - cos β sin α}{cos β cos α + sin β sin α}$ $(sinbetacosalpha-cosbetasinalpha)/(cosbetacosalpha+sinbetasinalpha)$

= $\frac{sin (β - α)}{cos (β - α)}$ $sin(beta-alpha)/cos(beta-alpha)$

= $tan (α - β)$ $tan(alpha-beta)$

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