Is csc^2(-θ)-1=cot^2(θ) ?

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

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prove csc^2(-θ)-1=cot^2(θ)

prove

csc^{2} (- θ) - 1 = cot^{2} (θ)

True

Solution steps

csc^{2} (- θ) - 1 = cot^{2} (θ)

Manipulating left side

csc^{2} (- θ) - 1

Use the negative angle identity:

csc (- x) = - csc (x)

= - 1 + (- csc (θ))^{2}

Simplify

= - 1 + csc^{2} (θ)

Rewrite using trig identities

- 1 + csc^{2} (θ)

Use the Pythagorean identity:

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

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

= cot^{2} (θ)

= cot^{2} (θ)

We showed that the two sides could take the same form

\Rightarrow True

\frac{( sec ( θ ) - tan ( θ ) ) ^{2} + 1}{sec ( θ ) csc ( θ ) - tan ( θ ) csc ( θ )} = 2 tan (θ)

\frac{1}{csc ( x ) - cot ( x )} = csc (x) + cot (x)

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

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

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