Question

a) cos⁴a - sin⁴a +1 = 2cos²a

b) cos⁶a + sin⁶a + 3sin²a.cos²a = 1

Answer 1

b: \(=\left(\cos^2\alpha+\sin^2\alpha\right)^3-3\cos^2\alpha\sin^2\alpha\left(\sin^2\alpha+\cos^2\alpha\right)+3\cdot\sin^2\alpha\cdot\cos^2\alpha\)

=1

Answer 2

\(cos^4a-sin^4a+1=\left(cos^2a-sin^2a\right)\left(cos^2a+sin^2a\right)+1\)

\(=cos^2a-sin^2a+1=cos^2a-sin^2a+sin^2a+cos^2a\)

\(=2cos^2a\)

\(cos^6a+sin^6a+3sin^2a.cos^2a\)

\(=\left(cos^2a+sin^2a\right)^3-3sin^2a.cos^2a\left(sin^2a+cos^2a\right)+3sin^2a.cos^2a\)

\(=1-3sin^2a.cos^2a.1+3sin^2a.cos^2a\)

\(=1\)