Question

if costeta - sinteta = root2sinteta, then show that costeta + sinteta = root2costeta​

Answer 1

Answer:

Cosθ + Sinθ = √2 Cosθ

Step-by-step explanation:

Given--->

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Cosθ - Sinθ = √2 Sinθ

To prove--->

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Cosθ + Sinθ = √2 Cosθ

Proof--->

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Cosθ - Sinθ = √2 Sinθ

=> Cosθ = √2 Sinθ + Sinθ

=> Cosθ = (√2 + 1) Sinθ

multiplying both sides by (√2 - 1)

=> (√2 - 1) Cosθ =(√2 - 1) (√2 + 1) Sinθ

We have an identity

a² - b² =(a + b ) (a - b )

=> (√2 - 1) Cosθ = {(√2)² - ( 1 )² } Sinθ

=> (√2 - 1)Cosθ = (2 - 1) Sinθ

=> √2Cosθ - Cosθ = Sinθ

=> √2Cosθ = Sinθ + Cosθ

=> Sinθ + Cosθ =√2 Cosθ

=> Cosθ + Sinθ = √2 Cosθ