Prove:(√3+1) (3 – cot 30°) = tan3 60° – 2 sin 60°

1.	Prove:(√3+1) (3 – cot 30°) = tan3 60° – 2 sin 60°
Answer» L.H.S: (√3 + 1) (3 – cot 30°) = (√3 + 1) (3 – √3) [∵cos 30° = √3] = (√3 + 1) √3 (√3 – 1) [∵(3 – √3) = √3 (√3 – 1)] = ((√3)²– 1) √3 [∵ (√3+1)(√3-1) = ((√3)² – 1)] = (3-1) √3 = 2√3 Similarly solving R.H.S: tan³ 60° – 2 sin 60° Since, tan 60^o = √3 and sin 60^o = √3/2, We get, (√3)³ – 2.(√3/2) = 3√3 – √3 = 2√3 Therefore, L.H.S = R.H.S Hence, proved.

Answer»

L.H.S: (√3 + 1) (3 – cot 30°)

= (√3 + 1) (3 – √3) [∵cos 30° = √3]

= (√3 + 1) √3 (√3 – 1) [∵(3 – √3) = √3 (√3 – 1)]

= ((√3)²– 1) √3 [∵ (√3+1)(√3-1) = ((√3)² – 1)]

= (3-1) √3

= 2√3

Similarly solving R.H.S: tan³ 60° – 2 sin 60°

Since, tan 60^o = √3 and sin 60^o = √3/2,

We get,

(√3)³ – 2.(√3/2) = 3√3 – √3

= 2√3

Therefore, L.H.S = R.H.S

Hence, proved.

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