Q15 of 268 Page 5

Prove the following identities :

2 cos2 θ – cos4 θ + sin4 θ = 1

Taking LHS = 2 cos2 θ – cos4 θ + sin4 θ

= 2 cos2 θ – (cos4 θ – sin4 θ)


= 2 cos2 θ – [(cos2 θ)2 – (sin2 θ)2]


Using identity, (a2 – b2) = (a + b) (a – b)


= 2 cos2 θ – [(cos2 θ – sin2 θ)(cos2 θ+ sin2 θ )]


= 2 cos2 θ – [(cos2 θ – sin2 θ)(1)] [∵ cos2 θ + sin2 θ = 1]


=2 cos2 θ – cos2 θ + sin2 θ


= cos2 θ + sin2 θ


= 1 [∵ cos2 θ + sin2 θ = 1]


= RHS


Hence Proved


More from this chapter

All 268 →
15

Prove the following identities :

sin4 A – cos4 A = 2 sin2 A – 1 = 1 – 2 cos2 A = sin2 A – cos2 A

 15

Prove the following identities :

cos4 θ – sin4 θ = cos2 θ – sin2 θ = 2 cos2 θ – 1

 15

Prove the following identities :

1 – 2 cos2 θ + cos4 θ = sin4θ

 15

Prove the following identities :

1 – 2 sin2 θ + sin4 θ = cos4θ