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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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Proof of: lim q →0 (cos q -1) / q = 0 We will begin by doing the following, lim q →0 (cosq -1)/q = lim q →0 ((cosq - 1)(cosq + 1))/(q (cosq + 1)) = lim q
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