Verify sin^4 x - sin^2 x = cos^4 x - cos^2 x is an identity.

Question

Mathematics

Gilberto Moreno

16 March, 09:18

Verify sin^4 x - sin^2 x = cos^4 x - cos^2 x is an identity.

+1

Answers (1)

Know the Answer?

Brooklyn Carpenter · Answer 1

(identity has been verified)

Step-by-step explanation:

Verify the following identity:

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

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

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

- (1 - cos (x) ^2) = cos (x) ^2 - 1:

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

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

-1 + cos (x) ^2 + (1 - cos (x) ^2) ^2 = ^? cos (x) ^4 - cos (x) ^2

(1 - cos (x) ^2) ^2 = 1 - 2 cos (x) ^2 + cos (x) ^4:

-1 + cos (x) ^2 + 1 - 2 cos (x) ^2 + cos (x) ^4 = ^? cos (x) ^4 - cos (x) ^2

-1 + cos (x) ^2 + 1 - 2 cos (x) ^2 + cos (x) ^4 = cos (x) ^4 - cos (x) ^2:

cos (x) ^4 - cos (x) ^2 = ^? cos (x) ^4 - cos (x) ^2

The left hand side and right hand side are identical:

Answer: (identity has been verified)