Question

prove that cot(x/2)-tan(x/2)=2cot(x)

Answer 1

Answer :

We have to prove,
cot(x/2)-tan(x/2)=2cot(x)
LHS = cot(x/2)-tan(x/2) & RHS = 2cot(x)

These formulas will be used in this question-
•cot x = cos x / sin x ------------(1)
•tanx = sin x / cos x -----------(2)
• cos(2x) = cos^(x) – sin^(x) ------(3)
•sin 2x = 2 sin x . cos x --------(4)

LHS = cot(x/2) - tan(x/2)
Using formula no (1) & (2) from above,
LHS = {cos(x/2)/sin(x/2)} - {sin(x/2)/cos(x/2)}

=> LHS = {cos^(x/2) - sin^(x/2)}/sin(x/2).cos(x/2)

Now using the formula no (3) from above,
LHS = cos(x) / sin(x/2).cos(x/2)

Lets multiply and divide the denominator by 2,
=> LHS = cos(x)/ {2sin(x/2).cos(x/2)}/2

Now using formula no (4) from above,
LHS = 2cos(x) / sin(x)
=> LHS = 2cot(x) = RHS

Hence its proved that,

●cot(x/2)-tan(x/2) = 2cotx

●●● Hope It Helps ●●●