Is 1-cot^2(θ)=2-csc^2(θ) ?

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

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prove 1-cot^2(θ)=2-csc^2(θ)

prove

1 - cot^{2} (θ) = 2 - csc^{2} (θ)

True

Solution steps

1 - cot^{2} (θ) = 2 - csc^{2} (θ)

Manipulating left side

1 - cot^{2} (θ)

Rewrite using trig identities

1 - cot^{2} (θ)

Use the Pythagorean identity:

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

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

= 1 - (csc^{2} (θ) - 1)

Simplify

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

1 - (csc^{2} (θ) - 1)

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

- (csc^{2} (θ) - 1)

Distribute parentheses

= - (csc^{2} (θ)) - (- 1)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - csc^{2} (θ) + 1

= 1 - csc^{2} (θ) + 1

Simplify

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

1 - csc^{2} (θ) + 1

Group like terms

= - csc^{2} (θ) + 1 + 1

Add the numbers:

1 + 1 = 2

= - csc^{2} (θ) + 2

= - csc^{2} (θ) + 2

= - csc^{2} (θ) + 2

= - csc^{2} (θ) + 2

= 2 - csc^{2} (θ)

We showed that the two sides could take the same form

\Rightarrow True