Question

cot x/2 - tan x/2 = 2cotx​

Answer 1

Step-by-step explanation:

Identity:tan (x/2)=(sin x)/(1+cos x)

cot (x/2)=1/tan (x/2)=(1+cos x)/(sin x)

..

1/2 cot x/2 - 1/2 tan x/2 = cot x

start with left side:

1/2 cot x/2 - 1/2 tan x/2

=(1+cos x)/2(sin x)-(sin x)/2(1+cos x)

LCD: 2sin x(1+cos x)

=(1+cos x)^2-sin^2x)/(2sin x(1+cos x)

=(1+2cos x+cos^2x-sin^2x)/(2sin x(1+cos x)

1-sin^2x=cos^2x

=(2cos x+2cos^2x)/(2sin x(1+cos x)

=(cos x+cos^2x)/(sin x(1+cos x)

=(cos x(1+cos x)/(sin x(1+cos x)

=cos x/sin x

=cot x

verified: left side=right side

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Answer 2

Step-by-step explanation:

Step-by-step explanation:

Identity:tan (x/2)=(sin x)/(1+cos x)

cot (x/2)=1/tan (x/2)=(1+cos x)/(sin x)

..

1/2 cot x/2 - 1/2 tan x/2 = cot x

start with left side:

1/2 cot x/2 - 1/2 tan x/2

=(1+cos x)/2(sin x)-(sin x)/2(1+cos x)

LCD: 2sin x(1+cos x)

=(1+cos x)^2-sin^2x)/(2sin x(1+cos x)

=(1+2cos x+cos^2x-sin^2x)/(2sin x(1+cos x)

1-sin^2x=cos^2x

=(2cos x+2cos^2x)/(2sin x(1+cos x)

=(cos x+cos^2x)/(sin x(1+cos x)

=(cos x(1+cos x)/(sin x(1+cos x)

=cos x/sin x

=cot x

__LHS=RHS