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

(cot (x) - csc (x)) (cos (x) + 1) = - sin (x)

tan (t) cot (t) = 1

csc^{4} (θ) - cot^{4} (θ) = 2 csc^{2} (θ) - 1

csc (x) - cos (x) cot (x) = sin (x)

\frac{csc ^{2} ( θ )}{cot ( θ )} = csc (θ) sec (θ)

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