Math, asked by Yashvardhan1818, 9 months ago

If x=cos55degree ,y=cos65degree,z=cos175degree then xy+yz+zx=?

Answers

Answered by kushalchauhan07
0

Answer:

xy + yz + zx =  cos55º \times cos65º + cos65º \times cos175º + cos175º \times cos55º

We know that 2cosAcosB = cos(A + B) + cos(A-B).

= > (1/2)(cos(55 + 65) + cos(65 - 55)) + (1/2)(cos(65 + 175) + cos(175 - 65)) + (1/2)(cos(175 + 55)cos(175 - 55))

= > (1/2)(cos120+cos10) + (1/2)(cos240+cos110) + (1/2)(cos230 + cos120)

= > (1/2)(cos120) + (1/2)cos(10) + (1/2)cos240 + (1/2)cos(110) + (1/2)cos(230) + (1/2)cos120

= > (1/2)cos(180 - 60) + (1/2)cos(10) + (1/2)cos(180 + 60) + (1/2)cos(110) + (1/2)(cos230) - 1/2(cos(180 - 60))

= > (1/2)(-cos60) + (1/2)cos(10) + (1/2)cos(-cos60) + (1/2)cos110 + (1/2)cos230 - 1/2(-cos60)

= > (1/2)(-1/2) + (1/2)cos(10) + (1/2)(-1/2) + (1/2)cos(110) + (1/2)cos(230) - (1/2)(-1/2)

= > -1/4 + 1/2cos(10) - 1/4 + (1/2)cos(110) + (1/2)cos230 - 1/4

= > -(3/4) + (1/2)(cos10 + cos110 + cos230)

</em></p><p><em>= &gt; \: -  \frac{3}{4}  + ( \frac{1}{2} )(  \frac{2cos(110 + 10)}{2}   \frac{2cos(110 - 10)}{2}  + cos230)

</em></p><p><em>= &gt; -  \frac{3}{4} +   \frac{1}{2} (2cos60cos50 + cos(180 + 50)

</em></p><p><em>= &gt; -  \frac{3}{4}  +  \frac{1}{2} (2cos60cos50 - cos50)

</em></p><p><em>= &gt;  - \frac{3}{4}  +  \frac{1}{2} cos50(2 \times  \frac{1}{2}  - 1)

</em></p><p><em>= &gt; -  \frac{3}{4}  +  \frac{1}{2} cos50(0)

</em></p><p><em>= &gt; -  \frac{3}{4}

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