Is 2-csc^2(θ)=1-cot^2(θ) ?

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

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prove 2-csc^2(θ)=1-cot^2(θ)

prove

2 - csc^{2} (θ) = 1 - cot^{2} (θ)

True

Solution steps

2 - csc^{2} (θ) = 1 - cot^{2} (θ)

Manipulating left side

2 - csc^{2} (θ)

Rewrite using trig identities

2 - csc^{2} (θ)

Use the Pythagorean identity:

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

= 2 - (1 + cot^{2} (θ))

Simplify

2 - (1 + cot^{2} (θ)) : - cot^{2} (θ) + 1

2 - (1 + cot^{2} (θ))

- (1 + cot^{2} (θ)) : - 1 - cot^{2} (θ)

- (1 + cot^{2} (θ))

Distribute parentheses

= - (1) - (cot^{2} (θ))

Apply minus-plus rules

+ (- a) = - a

= - 1 - cot^{2} (θ)

= 2 - 1 - cot^{2} (θ)

Subtract the numbers:

2 - 1 = 1

= - cot^{2} (θ) + 1

= - cot^{2} (θ) + 1

= - cot^{2} (θ) + 1

= 1 - cot^{2} (θ)

We showed that the two sides could take the same form

\Rightarrow True