Prove that cos - sin = v2 sin , Mathematics

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If cos?+sin? = √2 cos?, prove that cos? - sin? =  √2 sin ?.

Ans:    Cos? + Sin? =  √2 Cos?

⇒ ( Cos? + Sin?)2  = 2Cos2?

⇒ Cos2? + Sin2?+2Cos? Sin? = 2Cos2?

⇒ Cos2? - 2Cos? Sin?+ Sin2? = 2Sin2?                (  ∴2Sin2? = 2 - 2Cos2?)

⇒ (Cos? - Sin?)2  = 2Sin2?                                (1- Cos2? = Sin2? & 1 - Sin2? = Cos2?)

or Cos? - Sin? = √2  Sin?.

 


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