Prove that: (cos^3x - sin^3x)/(cosx - sinx) = 1 + cosx*sinx is always true.
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Tushar Chandra
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We have to prove that [(cos x)^3 - (sin x)^3]/[cos x - sin x]= 1 + cos x * sin x
Starting with the left hand side:
(cos x)^3 - (sin x)^3 / cos x - sin x
use a^3 - b^3 = (a - b)(a^2 + ab + b^2)
=> [(cos x - sin x)[(cos x)^2 + cos x * sin x + (sin x)^2]]/( cos x - sin x)
cancel (cos x - sin x)
=> [(cos x)^2 + cos x * sin x + (sin x)^2]
Use (cos x)^2 + (sin x)^2 = 1
=> 1 + cos x * sin x
This is the right hand side.
This proves that [(cos x)^3 - (sin x)^3]/[cos x - sin x] = 1 + cos x * sin x
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