Verify the identity sinx/(1-cosx)=(1+cosx)/sinx.
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We need to verify that sinx/(1-xosx)= (1+cosx)/sinx
We know that sin^2(x) + cos^2(x)=1
==> sin^2(x)= 1-cos^2(x)
==> sin^2(x)= (1-cos(x))(1+cos(x))
==> (sinx)(sinx)= (1-cosx)(1+cosx)
Now divide by (sinx)(1-cosx)
==> sinx/(1-cosx)=(1+cosx)/sinx)
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To verify whether sinx/(1-cosx) =(1+cosx)/sinx.
Solution:
We know that
sin^2x+cos^2x = 1. Or
sin^2x = 1-cos^2x = (1+cosx)(1-cosx). Dividing both sides by sinx(1-cosx) we get:
sinx/(1-cosx) = (1+cosx)/sinx
To verify the identity, first, we have to cross multiply, so that:
sinx*sinx = (1-cosx)*(1+cosx)
We notice that, to the right side, the product will become a difference of squares:
(sinx)^2 = 1 - (cosx)^2
We'll move the term - (cosx)^2, to the left side, changing it's sign and we'll get:
(sinx)^2 + (cosx)^2 = 1
The relation above is true, being the fundamental formula of trigonometry.
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