If A + B + C = π, prove that cos 2A – cos 2B + cos 2C = 1 – 4sin A cos B sin C

Question

Philoid · Accepted Answer

= cos2A – cos2B + cos2C

Using,

= cos2A - {2sin(B+C)sin(B-C)}

since A + B + C = π

And sin(π – A) = sinA

= cos2A – {2sin(π – A)sin(B-C)}

= cos2A – {2sinAsin(B-C)}

= cos2A - 2sinAsin(B-C)

Using , cos2A = 1 – 2sin²A

= -2sin²A + 1 – 2sinAsin(B-C)

= -2sinA{sinA + sin(B-C)} + 1

= -2sinA{2cosCsinB} + 1

= -4sinAcosBsinC + 1

= R.H.S