Q-1-1-cos-2-2A-2-cos-4-A-sin-4-A-2-sin-2-A-sin-2-120-0-A-sin-2-120-0-A-3-2-

Question Number 8401 by rhm last updated on 10/Oct/16

Q . 1 1 + {c o s}^{2} 2 A = 2 ({c o s}^{4} A + {s i n}^{4} A)

2 . {s i n}^{2} A + {s i n}^{2} (120^{0} + A) + {s i n}^{2} (120^{0} - A)

= \frac{3}{2}

Answered by ridwan balatif last updated on 10/Oct/16

Q . 1 1 + \cos^{2} 2 A = 2 (\cos^{4} A + \sin^{4} A)

Missing \left or extra \right

= 2 ({(\cos^{2} A + \sin^{2} A)}^{2} - {2 \sin}^{2} {Acos}^{2} A)

= 2 (1 - (2 sinAcosA) . \frac{2}{2} sinAcosA)

= 2 (1 - \frac{\sin^{2} 2 A}{2})

= 2 - \sin^{2} 2 A

= 1 + 1 - \sin^{2} 2 A

= 1 + \cos^{2} 2 A

∴ 1 + \cos 2 A = 2 (\cos^{4} A + \sin^{4} A) (PROVE)

Answered by ridwan balatif last updated on 10/Oct/16

2 . \sin^{2} A + {(\sin (120^{o} + A))}^{2} + {(\sin (120^{o} - A))}^{2}

\sin^{2} A + {({\sin 120}^{o} cosA + {sinAcos 120}^{o})}^{2} + {({\sin 120}^{0} cosA - {sinAcos 120}^{o})}^{2}

\sin^{2} A + {(\frac{1}{2} \sqrt{3} cosA - \frac{1}{2} sinA)}^{2} + {(\frac{1}{2} \sqrt{3} cosA + \frac{1}{2} sinA)}^{2}

\sin^{2} A + {(\frac{1}{2} \sqrt{3} cosA - \frac{1}{2} sinA) + (\frac{1}{2} \sqrt{3} cosA + \frac{1}{2} sinA)}^{2} - 2 (\frac{1}{2} \sqrt{3} cosA - \frac{1}{2} sinA) (\frac{1}{2} \sqrt{3} cosA + \frac{1}{2} sinA)

\sin^{2} A + {(\sqrt{3} cosA)}^{2} - 2 (\frac{3}{4} \cos^{2} A - \frac{1}{4} \sin^{2} A)

\frac{3}{2} \sin^{2} A + \frac{3}{2} \cos^{2} A

\frac{3}{2} (\sin^{2} A + \cos^{2} A)

\frac{3}{2}

∴ \sin^{2} A + \sin^{2} (120^{o} + A) + \sin^{2} (120^{o} - A) = \frac{3}{2}