Question

Prove the following:​

Answer 1

✪QUESTION: - Prove that Sin^2 18° + cos^2 36° = 3/4.

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✍PROOF :-

Let,

L.H.S. = sin² 18 + cos² 36

=> sin² 18 + ( 1 - 2sin² 18)² - - - - - - (I)

Now, sin 36 = cos 54 - - {compl. rule }

=> sin 2A = cos 3A

=> 2 sin A cos A = 4 cos^3 A - 3 cos A

Dividing by cosA,

=> 2 sin A = 4 cos² A - 3

=> 2 sin A = 4 - 4 sin² A - 3

=> 4 sin² A + 2 sin A - 1 = 0

=> 4 sin² (18) + 2 sin (18) - 1 = 0

By quadratic formula,

=> sin 18 = [-2 ± √(4 + 16) ] / 8

=> sin 18 = (√5 - 1)/4

=> sin² 18 = [5 + 1 - 2√5] /16

=> sin² 18 = (3 - √5)/8

Substituting this in - - (i),

sin² 18 + ( 1 - 2sin² 18)² = (3 - √5)/8 + [1 - 2 (3 - √5)/8]

=> (3 - √5)/8 + [1 - (3 - √5)/4]²

=> (3 - √5)/8 + [(4 - 3 + √5)/4]²

=> (3 - √5)/8 + [(1 + √5)/4]²

=> (3 - √5)/8 + (1+ 5+ 2√5)/16

=> (3 - √5)/8 + (3 + √5)/8

=> (3 - √5 + 3 + √5)/8

=> 3/4 = RHS.

HENCE PROVED

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HOPE IT HELPS :)