Consider the following: 1. \cos^6A+\sin^6A=1-3\sin^2A\cos^2A 2. \cos^4A+\sin^4A=1-2\sin^2A\cos^2A 3. \cos^4A+\sin^4A=2\cos^2A-1 Which of the statements given above are correct?

Trigonometry Practice Test 1

Question : 16 of 40

Marks: +1, -0

Consider the following:
1.

\cos^6A+\sin^6A=1-3\sin^2A\cos^2A

\cos^4A+\sin^4A=1-2\sin^2A\cos^2A

\cos^4A+\sin^4A=2\cos^2A-1

Which of the statements given above are correct?

Solution:

\cos^6A+\sin^6A= (\cos^2A)^3+( \sin^2A)^3

=(\cos^2A+\sin^2A)(\cos^4A+\sin^4A-\sin^2A\cos^2A)

=[(\cos^2A+\sin^2A)^2-2\sin^2A\cos^2A-\sin^2A\cos^2A]

⇒\cos^6A+\sin^6A=1-3\sin^2A\cos^2A

\cos^4A+\sin^4A= (\cos^2A+\sin^2A)^2-2\sin^2A\cos^2A

=1-2\sin^2A\cos^2A

\cos^4A+\sin^4A= (\cos^2A)^2-(\sin^2A)^2

=(\cos^2A-\sin^2A)(\cos^2A+\sin^2A)

=\cos2A=2\cos^2A-1

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