Question

Verify the following trigonometric identity: (csc x + cot x)^2 = cos x +1/ 1- cos x

Answer 1

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Explanation:

As written, the equation is not an identity. Perhaps you want to show ...

  (csc(x) +cot(x))² = (cos(x) +1)/(1 -cos(x))

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We will transform the left-side expression to the form of the right-side expression.

  $(\csc(x)+\cot(x))^2=\left(\dfrac{1}{\sin(x)}+\dfrac{\cos(x)}{\sin(x)}\right)^2=\dfrac{(1+\cos(x))^2}{\sin(x)^2}\\\\=\dfrac{(1+\cos(x))^2}{1-\cos(x)^2}=\dfrac{1+\cos(x)}{1-\cos(x)}\cdot\dfrac{1+\cos(x)}{1+\cos(x)}=\boxed{\dfrac{\cos(x)+1}{1-\cos(x)}}$