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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Expand

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

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

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

= csc^{2} (x) - 1

Use the Pythagorean identity:

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

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

= cot^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

tan^{2} (y) (csc^{2} (y) - 1) = 1

2 sin (x) cos (x) + 1 = (sin (x) + cos (x))^{2}

1 - cos (θ) = sin (θ)

cos (2 π + x) = cos (x)

sin^{2} (x) sec^{2} (x) = \frac{1}{cot ^{2} ( x )}

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