If sin ( θ + α ) = a & sin ( θ + β ) = b (0 < α , β , θ < π /2) then find the value of cos2 ( α − β ) − 4 abcos( α − β )
Text Solution
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If a = sin( θ + α ), b = sin( θ + β )
∴ 2ab = 2sin ( θ + α ) sin ( θ + β )
2ab = cos ( α – β ) – cos (2 θ + α + β )
Multiply both sides by 2cos( α – β ) ⇒ 4ab cos ( α – β ) = 2cos 2 ( α – β ) – 2cos (2 θ + α + β ) . cos ( α – β )
= 1 + cos2 ( α – β ) – cos2( θ + α ) – cos2 ( θ + β )
⇒ cos2( α – β ) – 4ab cos ( α – β ) = cos2( θ + α ) + cos2 ( θ + β ) – 1
= 1 – 2sin 2 ( θ + α ) + 1 –2sin2( θ + β ) – 1
= 1 – 2a 2 – 2b 2
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