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.

Prove the identity cos4θ=18sin2θcos2θ.

L.H.S=cos4θ=cos2(2θ)=12sin22θ=12(2sinθcosθ)2=18sin2θcos2θ=R.H.S.

Prove the identity sin6xsinx+cos4xcos3x=cos3xcos2x.

L.H.S.=sin6xsinx+cos4xcos3x=12(2sin6xsinx+2cos4xcos3x)=12[cos(6xx)cos(6x+x)+(cos(4x+3x)+cos(4x3x))]=12[cos5xcos7x+(cos7x+cosx)]=12[cos5xcos7x+cos7x+cosx]=12[cos5x+cosx]=12[2cos5x+x2+cos5xx2]=12(2cos3x+cos2x)cos3x+cos2x=R.H.S.