# cos^2(36) + sin^2(18)

• Sep 24th 2011, 02:12 AM
afrim1
cos^2(36) + sin^2(18)
can you walidate ?

cos²(36°) + sin²(18°) = 3/4
• Sep 24th 2011, 04:04 AM
issacnewton
Re: cos^2(36) + sin^2(18)
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 - - {Where A = 18 }
=> 2 sin A cos A = 4 cos^3 A - 3 cos A
=> 2 sin A = 4 cos² A - 3 - - - {Divide by cos A }
=> 2 sin A = 4 - 4 sin² A - 3
=> 4 sin² A + 2 sin A - 1 = 0
=> 4 sin² (18) + 2 sin (18) - 1 = 0

=> sin 18 = [-2 ± √(4 + 16) ] / 8 - - - - { By Quadratic Formula }
=> sin 18 = (√5 - 1)/4
=> sin² 18 = [5 + 1 - 2√5] /16
=> sin² 18 = (3 - √5)/8

Substituting this in - - (i), we get,

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

reference--- Prove that sin^2 18 + cos^2 36 = 3/4? - Yahoo! Answers India