Prove that............
Answers
Answer:
Multiply numerator and denominator by (cosec θ + cot θ).
Then the numerator is (cosec θ + cot θ)², which is what we want to finish with, so the denominator had better be equal to 1.
The denomiator is then
(cosec θ + cot θ)(cosec θ - cot θ)
= cosec² θ - cot² θ
= 1 / sin² θ - cos² θ / sin² θ
= ( 1 - cos² θ ) / sin² θ
= sin² θ / sin² θ
= 1.
That finishes the proof of the first equation.
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Next
(cosec θ + cot θ)²
= cosec² θ + cot² θ + 2 cosec θ cot θ
= 1 / sin² θ + cot² θ + 2 cosec θ cot θ
= ( sin² θ + cos² θ ) / sin² θ + cot² θ + 2 cosec θ cot θ
= 1 + cot² θ + cot² θ + 2 cosec θ cot θ
= 1 + 2 cot² θ + 2 cosec θ cot θ
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Step-by-step explanation:
Multiply numerator and denominator by (cosec θ + cot θ).
Then the numerator is (cosec θ + cot θ)², which is what we want to finish with, so the denominator had better be equal to 1.
The denomiator is then
(cosec θ + cot θ)(cosec θ - cot θ)
= cosec² θ - cot² θ
= 1 / sin² θ - cos² θ / sin² θ
= ( 1 - cos² θ ) / sin² θ
= sin² θ / sin² θ
= 1.
That finishes the proof of the first equation.
----------------
Next
(cosec θ + cot θ)²
= cosec² θ + cot² θ + 2 cosec θ cot θ
= 1 / sin² θ + cot² θ + 2 cosec θ cot θ
= ( sin² θ + cos² θ ) / sin² θ + cot² θ + 2 cosec θ cot θ
= 1 + cot² θ + cot² θ + 2 cosec θ cot θ
= 1 + 2 cot² θ + 2 cosec θ cot θ