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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