# trignometric identities: prove 1/sinθ-sinθ≡cosθ/tanθ

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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**