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We need to prove that :
1/sin(x) - sinx/tanx = cosx/ tanx
We will start from the right side and prove the left side.
==> cos x/ tanx
But we know that tan(x)= sinx / cosx
We willl substitute:
==> cosx / (sinx/cosx) = cosx* cosx/ sinx = cos^2 x / sinx
But we know that cos^2 x = 1- sin^2 x
==> (1- sin^2 x) / sinx = 1/sinx - sin^2 x/ sinx = 1/sinx - sinx
==> Then we prove that cosx/ tanx = 1/sinx - sinx
We have to prove that 1/sin θ - sin θ = cos θ / tan θ is an identity.
Start with the left hand side:
1/sin θ - sin θ
=> 1/sin θ - (sin θ)^2/sin θ
=> [1 - (sin θ)^2]/sin θ
=> (cos θ)^2/sin θ
=> (cos θ)/(sin θ/cos θ)
=> cos θ/tan θ
which is the right hand side.
This proves that 1/sin θ - sin θ = cos θ / tan θ is an identity
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