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

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

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

prove

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

True

Solution steps

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

Manipulating left side

(csc (x) - cot (x)) (csc (x) + cot (x))

Expand

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

(csc (x) + cot (x)) (csc (x) - cot (x))

Apply Difference of Two Squares Formula:

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

a = csc (x), b = cot (x)

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

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

Use the Pythagorean identity:

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

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

= 1

We showed that the two sides could take the same form

\Rightarrow True

(tan (x)) (sin (x)) = sec (x) - cos (x)

tan (α) + cot (α) = sec (α) csc (α)

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

\frac{cos ( x + y )}{cos ( x - y )} = \frac{cot ( y ) - tan ( x )}{cot ( y ) + tan ( x )}

cot (x) tan (x) = 1

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