Question

show that cos 3 theta - sin 3 theta = (cos theta + sin theta)(1-sin 2 theta)​

Answer 1

cos3Θ - sin3Θ

=4cos³Θ - 3cosΘ - (3sinΘ - 4sin³Θ)

=4cos³Θ - 3cosΘ - 3sinΘ + 4sin³Θ

=4(cos³Θ + sin³Θ) -3(cosΘ + sinΘ)

=4(cosΘ + sinΘ)(cos²Θ - sinΘcosΘ + sin²Θ) - 3(cosΘ + sinΘ)

=4(cosΘ + sinΘ)(1 - sinΘcosΘ) - 3(cosΘ + sinΘ)

=(cosΘ + sinΘ)(4 - 4sinΘcosΘ -3)

=(cosΘ + sinΘ)(1 - 2sin2Θ)

∴HENCE PROVED