Simplify the expression.
(1 + cot θ)(1 - cot θ) - csc^2 θ
(1 + cot θ)(1 - cot θ) - csc^2 θ
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(1 + cot θ)(1 - cot θ) - csc^2 θ
= (1 - cot^2 θ) - csc^2 θ (using the identity (x - y)(x + y) = x^2 - y^2)
= (csc^2 θ - 2cot^2 θ) - csc^2 θ (since csc^2 θ - cot^2 θ = 1)
= -2cot^2 θ.
= (1 - cot^2 θ) - csc^2 θ (using the identity (x - y)(x + y) = x^2 - y^2)
= (csc^2 θ - 2cot^2 θ) - csc^2 θ (since csc^2 θ - cot^2 θ = 1)
= -2cot^2 θ.
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(1 + cot θ)(1 - cot θ) - csc^2 (θ) [given]
1 - cot^2 (θ) - csc^2 (θ) [FOIL]
There's multiple ways to approach this from here; this is the first method:
1 - cot^2 (θ) - (1 + cot^2 (θ)) [using the pythagorean identity --> 1 + cot ²θ = csc²θ]
1 - cot^2 (θ) - 1 - cot^2 (θ) [distribute the negative sign]
- 2cot^2 (θ) [combine like terms]
Second method:
1 - (csc^2 (θ) - 1) - csc^2 (θ) [same pythagorean identity]
1 - csc^2 (θ) + 1 - csc^2 (θ) [distribute the negative sign]
2 - 2csc^2 (θ) [combine like terms]
2 (1 - csc^2 (θ)) [factor out the greatest common factor - 2]
1 - cot^2 (θ) - csc^2 (θ) [FOIL]
There's multiple ways to approach this from here; this is the first method:
1 - cot^2 (θ) - (1 + cot^2 (θ)) [using the pythagorean identity --> 1 + cot ²θ = csc²θ]
1 - cot^2 (θ) - 1 - cot^2 (θ) [distribute the negative sign]
- 2cot^2 (θ) [combine like terms]
Second method:
1 - (csc^2 (θ) - 1) - csc^2 (θ) [same pythagorean identity]
1 - csc^2 (θ) + 1 - csc^2 (θ) [distribute the negative sign]
2 - 2csc^2 (θ) [combine like terms]
2 (1 - csc^2 (θ)) [factor out the greatest common factor - 2]
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yeahh babyy