# Verify if cos^4x + 2sin^2x - sin^4x = 1.

justaguide | Certified Educator

calendarEducator since 2010

starTop subjects are Math, Science, and Business

We have to verify if cos^4x + 2sin^2x - sin^4x = 1.

Now cos^4x + 2sin^2x - sin^4x

=> [(cos x)^2]^2 + 2sin^2x - sin^4x

Now use the relation (cos x)^2 = 1 - (sin x)^2

=> [1 - (sin x)^2]^2 + 2sin^2x - sin^4x

Open the brackets

=> 1 + (sin x)^4 - 2( sin x)^2 +2sin^2x - sin^4x

cancel the common terms

=> 1

Therefore we have proved that cos^4x + 2sin^2x - sin^4x = 1

check Approved by eNotes Editorial

neela | Student

To verify  if cos^4x + 2sin^2x - sin^4x = 1.

We know that LHS : cos^4- - sin^4x = (cos^2x-sin^2x)(cos^2+sin^2x) = cos^2x-sin^2x, as cos^2+sin^2x = 1.

Therefore cos^4x-sin^4 = cos^2x-sin^2x.

cos^4x-sin^4x =   (1-sin^2) -sin^2x , as ccos^2 = 1-sin^2x

=> cos^4x- sin^4x = 1-2sin^2x.

We add 2sin^2sx to both sides and get:

cos^4x+2sin^2-sin^4 x =  1-2sin^2x +2sin^2x = 1.

Therefore cos^4x+2sin^2-sin^4x = 1.

check Approved by eNotes Editorial
giorgiana1976 | Student

We'll re-write the identity, so that to get both sides differences of squares:

(cos x)^4 - (sin x)^4 = 1 - 2(sin x)^2

We'll write the difference of square from the left side, using the formula:

a^2 - b^2 = (a-b)(a+b)

(cos a)^4 - (sin a)^4 = [(cos a)^2 - (sin a)^2][(cos a)^2 + (sin a)^2]

We'll recall the fundamental formula of trigonometry:

(cos a)^2 + (sin a)^2 = 1

(cos a)^4 - (sin a)^4 = [(cos a)^2 - (sin a)^2]

We'll substitute (cos a)^2 = 1 - (sin a)^2

(cos a)^4 - (sin a)^4 = 1 - (sin a)^2 - (sin a)^2

We'll combine like terms:

(cos a)^4 - (sin a)^4 = 1 - 2(sin a)^2 q.e.d

We notice that it's no need to transform the right side, since working to the left side, we've get the same expression we have to the right side.

check Approved by eNotes Editorial