Question

if y= cot ^-1 [√(cosx) - tan^-1[√(cosx) then P.T. sin y= tan^2(x/2)

Answer 1

y= cot-1√(cosx) - tan-1√(cosx)

applying sin on both sides we get,

siny= sin{tan-11/ √(cosx) - tan-1(cosx)}

siny= sin{tan-1(1-cosx/2√(cosx))}

siny= sin{sin-1(1-cosx/1+cosx)}

siny= 1-cosx/1+cosx

we know that 1-cosx/1+cosx= (tanx)^2/2

therefore,

siny=(tanx)^2/2

hence proved