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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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