Question

If y = cot-1(sq root cosx) - tan-1(sq root cosx) the prove that siny = tan2(x/2)

Answer 1

y= cot-¹( √cosx ) - tan-¹( √cosx)
Let cot-¹(√cosx) =P
cotP = √cosx
sinP = 1/√(1 + cosx)
P = sin-¹{1/√(1 + cosx)}


Let tan-¹( √cosx) = Q
tanQ = √cosx
sinQ = √cosx/√(1 + cosx)
Q = sin-¹( √cosx/√(1 + cosx )

y = P - Q
siny = sin(P - Q )
= sinP.cosQ - cosP.sinQ
= 1/√(1 + cosx).1/√(1 + cosx) - √cosx/√(1 + cosx) .√cosx/√(1 + cosx)
= (1 - cosx)/(1 + cosx)
= 2sin²x/2/2cos²x/2
= tan²x/2
Hence, siny = tan²x/2

Answer 2

Answer:

Firstly let sin inverse 3/4 be theta then it will become

Tan theta/2 and we know that it is equal to whole under root 1-cos theta / 1+ cos theta now find cos theta using the value of sin theta which is 3/4 then cos theta = √7/4 put the value in whole under root formula and get the answer : ) Hope it helped