cot x/2 - tan x/2 = 2cotx
Answers
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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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