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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Use the negative angle identity:

csc (- x) = - csc (x)

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

Rewrite using trig identities

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

Expand

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

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

Apply Difference of Two Squares Formula:

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

a = 1, b = csc (x)

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

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - csc^{2} (x)

= 1 - csc^{2} (x)

Use the Pythagorean identity:

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

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

= - cot^{2} (x)

= - cot^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

cos (0) - sec (0) = - sin (0) tan (0)

cot^{2} (x) sin^{2} (x) csc^{2} (x) tan^{2} (x) = 1

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

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

\frac{csc ( θ ) - sin ( θ )}{csc ( θ )} = cos^{2} (θ)

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