LE7Prove that : 4 sin 275V5) (3-v5).

1.	LE7Prove that : 4 sin 275V5) (3-v5).
Answer» We know thatSin 54° = Cos 36° = ( 1/4 ) (√5 + 1) => Cos 54° = √( 1 - Sin² 54°) = ( 1/4 ) √[ 10 – 2 √5 ] => 1 – 2 Sin² 27° = ( 1/4 ) √[ 10 – 2 √5 ] => 2 Sin² 27° = 1/2 [ 1 - ( 1/4 ) √[ 10 – 2 √5 ] Multiply both the sides by 8 – => 16 Sin² 27° = 8 - 2 √[ 10 – 2 √5 ] = [√ ( 5 + √5 ) - √ ( 3 - √5 ) ] ² => 4 Sin 27° = √( 5 + √5 ) - √( 3 - √5 ) ......... Proved (Neglecting – ve value as Sin is + ve.

Answer»

We know thatSin 54° = Cos 36° = ( 1/4 ) (√5 + 1)

=> Cos 54° = √( 1 - Sin² 54°) = ( 1/4 ) √[ 10 – 2 √5 ]

=> 1 – 2 Sin² 27° = ( 1/4 ) √[ 10 – 2 √5 ]

=> 2 Sin² 27° = 1/2 [ 1 - ( 1/4 ) √[ 10 – 2 √5 ]

Multiply both the sides by 8 –

=> 16 Sin² 27° = 8 - 2 √[ 10 – 2 √5 ] = [√ ( 5 + √5 ) - √ ( 3 - √5 ) ] ²

=> 4 Sin 27° = √( 5 + √5 ) - √( 3 - √5 ) ......... Proved

(Neglecting – ve value as Sin is + ve.

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