Is 1+cot^2(-θ)=csc^2(θ) ?

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

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prove 1+cot^2(-θ)=csc^2(θ)

prove

1 + cot^{2} (- θ) = csc^{2} (θ)

True

Solution steps

1 + cot^{2} (- θ) = csc^{2} (θ)

Manipulating left side

1 + cot^{2} (- θ)

Use the negative angle identity:

cot (- x) = - cot (x)

= 1 + (- cot (θ))^{2}

Simplify

= 1 + cot^{2} (θ)

Rewrite using trig identities

1 + cot^{2} (θ)

Use the Pythagorean identity:

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

= csc^{2} (θ)

= csc^{2} (θ)

We showed that the two sides could take the same form

\Rightarrow True

sec^{2} (x) = 1 + tan^{2} (x)

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

cos (x) (1 + tan^{2} (x)) = sec (x)

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

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