Derive a formula for tanx2 in terms of cosa Find the exact v

Derive a formula for tan(x/2) in terms of cos(a). Find the exact value of cos(37.5) and simplify completely. If cos(a + b) = 0, 0

4) tan(x/2) in terms of cosx

cosx = 2cos^2(x/2) -1

cos(x/2) = sqrt[(1+cosx)/2]

cosx = 1- 2sin^2(x/2)

sin(x/2) = sqrt[(1- cosx)/2]

So, tan(x/2) = sin(x/2) /cos(x/2) = sqrt{ [(1- cosx)/(1+cosx)]

5) cos(37.5)

cos(37.5) = cos(75/2) = sqrt[ (1+cos75)/2]

cos75 = cos(45 +30) = cos45cos30 - sin45sin30

=(1/sqrt2)(sqrt3/2) - (1/sqrt2)(1/2)

= ( sqrt3 -1)/2sqrt2

So, cos(37.5) = cos(75/2) = sqrt[ (1+cos75)/2]

= sqrt[ ( 1+ ( sqrt3 -1)/2sqrt2)/2 ]

= sqrt[ (4sqrt2 + sqrt3 -1)/sqrt2]/2