Question #058c3

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Answer 1 · 2018-02-17T04:11:09

# tan(a/2)#

# = sin(a/2)/cos(a/2)# ---(1)

Using the identities

# sin(a/2) = sqrt[(1 - cosa)/2]#

And

# cos(a/2) = sqrt[(1 + cosa)/2]#

Put these values in (1)

We get , # tan(a/2) #

#= sqrt[(1 - cosa)/2]/ sqrt[(1 + cosa)/2]#

#= sqrt[(1 - cosa)/(1 + cosa)]#

Multiply both numerator and denominator by #1 - cosa#

#= sqrt[(1-cosa)/(1+cosa) times (1-cosa)/(1-cosa)]#

#= sqrt[(1-cosa)^2 /(1-cos^2a)]#

We know , #1 - cos^2a = sin^2a#

#= sqrt[(1-cosa)^2 / sin^2a#

#= sqrt[((1-cosa)/sina)^2]#

#= (1-cosa)/sina#

#= 1/sina - cosa/sina#

#= csc a - cota#