Is csc^2(x)-1=(csc(x)-1)(csc(x)+1) ?

The answer to whether csc^2(x)-1=(csc(x)-1)(csc(x)+1) is True

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prove csc^2(x)-1=(csc(x)-1)(csc(x)+1)

prove

csc^{2} (x) - 1 = (csc (x) - 1) (csc (x) + 1)

True

Solution steps

csc^{2} (x) - 1 = (csc (x) - 1) (csc (x) + 1)

Manipulating right side

(csc (x) - 1) (csc (x) + 1)

Expand

(- 1 + csc (x)) (1 + csc (x)) : csc^{2} (x) - 1

(- 1 + csc (x)) (1 + csc (x))

Apply Difference of Two Squares Formula:

(a - b) (a + b) = a^{2} - b^{2}

a = csc (x), b = 1

= csc^{2} (x) - 1^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= csc^{2} (x) - 1

= - 1 + csc^{2} (x)

= csc^{2} (x) - 1

We showed that the two sides could take the same form

\Rightarrow True

Is csc^2(x)-1=(csc(x)-1)(csc(x)+1) ?
The answer to whether csc^2(x)-1=(csc(x)-1)(csc(x)+1) is True