Prove that: 20^ 100^ + 100^ 140^ - 140^ 200^ =-3 4 - Vidyadip

Question

Prove that: $\cos20^\circ\cos100^\circ+\cos100^\circ\cos140^\circ-\cos140^\circ\cos200^\circ=-\frac{3}{4}$

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Answer

We have, $\text{LHS}=\cos20^\circ\cos100^\circ+\cos100^\circ\cos140^\circ-\cos140^\circ\cos200^\circ$ $=\ \frac{1}{2}[2\cos100^\circ\cos20^\circ+2\cos140^\circ\cos100^\circ-2\cos200^\circ\cos140^\circ]$ $=\ \frac{1}{2}\big[\cos(100^\circ+20^\circ)+\cos(100^\circ-20^\circ)+\cos(140^\circ+100^\circ)\\ \ \ \ \ \ \ \ \ +\cos(140^\circ-100^\circ) -(\cos(200^\circ+140^\circ)+\cos(200^\circ-140^\circ))\big]$ $=\ \frac{1}{2}\big[\cos120^\circ+\cos80^\circ+\cos240^\circ+\cos40^\circ-\cos340^\circ-\cos60^\circ\big]$ $=\ \frac{1}{2}\Big[\cos(90^\circ+30^\circ)+\cos80^\circ+\cos40^\circ-\cos(180^\circ60^\circ)-\cos(360^\circ-20^\circ)-\frac{1}{2}\Big]$ $=\ \frac{1}{2}\Big[-\sin30^\circ+2\cos\Big(\frac{80^\circ+40^\circ}{2}\Big)\cos\Big(\frac{80^\circ-40^\circ}{2}\Big)-\cos60^\circ-\cos20^\circ-\frac{1}{2}\Big]$ $=\ \frac{1}{2}\Big[-\frac{1}{2}+2\cos60^\circ\cos20^\circ-\frac{1}{2}-\cos20^\circ-\frac{1}{2}$ $=\ \frac{1}{2}\Big[-\frac{3}{2}+2\times\frac{1}{2}\times\cos20^\circ-\cos20^\circ\Big]$ $=\ \frac{1}{2}\Big[-\frac{3}{2}+\cos20^\circ-\cos20^\circ\Big]$ $=\ \frac{1}{2}\Big[-\frac{3}{2}+0\Big]$ $=\ -\frac{3}{4}$ $=\ \text{RHS}$ $\therefore\ \cos20^\circ\cos100^\circ+\cos100^\circ\cos140^\circ-\cos140^\circ\cos200^\circ=-\frac{3}{4}.$ Hence proved.

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