Trigonometric Ratios
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 Trigonometric Ratio's: Proving the Identities Example: Prove the following identities: 2 (sin6 + cos6 ) - 3 (sin4 + cos4 ) + 1 = 0 Solution: We have, LHS = 2 (sin6 + cos6 ) - 3 (sin4 + cos4 ) + 1         = 2 [ (sin2 )3 + (cos2 )3 ] - [3 (sin2 )2 + (cos2 )2 ] + 1         = 2 [ (sin2 + cos2 ) {(sin2 )2 + (cos2 )2 - sin2 cos2 }] - 3 [(sin2 )2 + (cos2 )2 + 2 sin2 cos2 - 2 sin2 cos2 ] + 1         = 2 [ (sin2 )2 + (cos2 )2 + 2 sin2 cos2 - 3 sin2 cos2 ] - 3 [ (sin2 + cos2 )2 - 2 sin2 cos2 ] + 1         = 2 [ (sin2 + cos2 )2 - 3 sin2 cos2 ] - 3 [ 1 - 2 sin2 cos2 ] + 1         = 2 (1 - 3 sin2 cos2 ) - 3 (1 - 2 sin2 cos2 ) + 1         = 2 - 6 sin2 cos2 - 3 + 6 sin2 cos2 + 1   = 0         = 0 = RHS
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