SOLUTION: ((cos x)/(1-tan x))+((sin x)/(1-cot x))= sin x+ cos x How do you work out this identity?
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-> SOLUTION: ((cos x)/(1-tan x))+((sin x)/(1-cot x))= sin x+ cos x How do you work out this identity?
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Question 602462
:
((cos x)/(1-tan x))+((sin x)/(1-cot x))= sin x+ cos x
How do you work out this identity?
Answer by
jim_thompson5910(35256)
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cos(x)/(1 - tan(x))+sin(x)/(1 - cot(x)) = sin(x) + cos(x)
cos(x)/(1 - tan(x))+sin(x)/(1 - 1/tan(x)) = sin(x) + cos(x)
cos(x)/(1 - tan(x))+sin(x)/(tan(x)/tan(x) - 1/tan(x)) = sin(x) + cos(x)
cos(x)/(1 - tan(x))+sin(x)/( (tan(x)-1)/tan(x) ) = sin(x) + cos(x)
cos(x)/(1 - tan(x))+sin(x)*( tan(x)/(tan(x)-1) ) = sin(x) + cos(x)
cos(x)/(1 - tan(x))+(sin(x)*tan(x))/( tan(x)-1) = sin(x) + cos(x)
cos(x)/(1 - tan(x))+(sin(x)*tan(x))/( -(-tan(x)+1) ) = sin(x) + cos(x)
cos(x)/(1 - tan(x))+(sin(x)*tan(x))/( -(1 - tan(x)) ) = sin(x) + cos(x)
cos(x)/(1 - tan(x)) - (sin(x)*tan(x))/(1 - tan(x)) = sin(x) + cos(x)
(cos(x) - sin(x)*tan(x))/(1 - tan(x)) = sin(x) + cos(x)
(cos(x) - sin(x)*(sin(x)/cos(x)))/(1 - tan(x)) = sin(x) + cos(x)
(cos(x) - sin^2(x)/cos(x))/(1 - tan(x)) = sin(x) + cos(x)
(cos^2(x)/cos(x) - sin^2(x)/cos(x))/(1 - tan(x)) = sin(x) + cos(x)
((cos^2(x) - sin^2(x))/cos(x))/(1 - tan(x)) = sin(x) + cos(x)
((cos^2(x) - sin^2(x))/cos(x))*(1/(1 - tan(x))) = sin(x) + cos(x)
(cos^2(x) - sin^2(x))/(cos(x)(1 - tan(x))) = sin(x) + cos(x)
(cos^2(x) - sin^2(x))/(cos(x) - cos*tan(x)) = sin(x) + cos(x)
(cos^2(x) - sin^2(x))/(cos(x) - cos(x)*sin(x)/cos(x)) = sin(x) + cos(x)
(cos^2(x) - sin^2(x))/(cos(x) - sin(x)) = sin(x) + cos(x)
((cos(x) - sin(x))(cos(x) + sin(x)))/(cos(x) - sin(x)) = sin(x) + cos(x)
cos(x) + sin(x) = sin(x) + cos(x)
sin(x) + cos(x) = sin(x) + cos(x)
This verifies the identity.
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