Question 6 & 7, Review Exercise 10
Solutions of Question 6 & 7 of Review Exercise 10 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.
Question 6
Prove the identity cos4θ=1−8sin2θcos2θ.
Solution
L.H.S=cos4θ=cos2(2θ)=1−2sin22θ=1−2(2sinθcosθ)2=1−8sin2θcos2θ=R.H.S.
Question 7
Prove the identity sin6xsinx+cos4xcos3x=cos3xcos2x.
Solution
L.H.S.=sin6xsinx+cos4xcos3x=12(2sin6xsinx+2cos4xcos3x)=12[cos(6x−x)−cos(6x+x)+(cos(4x+3x)+cos(4x−3x))]=12[cos5x−cos7x+(cos7x+cosx)]=12[cos5x−cos7x+cos7x+cosx]=12[cos5x+cosx]=12[2cos5x+x2+cos5x−x2]=12(2cos3x+cos2x)cos3x+cos2x=R.H.S.
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