Question

Prove that cos55+cos65+cos175=0

Answer 1

$\huge\bigstar\mathfrak\green{\underline{\underline{SOLUTION:}}}$

L.H.S

cos55° + cos65° + cos175°

$(</strong><strong>cosA</strong><strong>+ </strong><strong>cosB</strong><strong> = 2cos \frac{</strong><strong>A</strong><strong> + </strong><strong>B</strong><strong>}{2} cos \frac{</strong><strong>A</strong><strong> </strong><strong>-</strong><strong>B</strong><strong>}{2} ) \\ \\ = > 2cos \frac{55 \degree + 65 \degree}{2} cos \frac{55 \degree - 65 \degree}{2} + cos(180 \degree - 5 \degree) \\ \\ = > 2cos \frac{120 \degree}{2} cos \frac{ - 10 \degree}{2} - cos5 \degree \\ \\ </strong><strong>[</strong><strong>cos</strong><strong>(180 \degree - </strong><strong>A</strong><strong>) = - </strong><strong>cosA</strong><strong>]</strong><strong> \\ \\ = > 2cos60 \degree \: cos( - 5 \degree) - cos5 \degree \\ \\ </strong><strong>[</strong><strong>cos</strong><strong>( - </strong><strong>A</strong><strong>) = </strong><strong>cosA</strong><strong>]</strong><strong> \\ \\ = > 2 \times \frac{1}{2} \times cos5 \degree - cos5 \degree \\ \\ = > cos5 \degree -cos5 \degree \\ \\ = > 0$

R.H.S

Hence proved ☺️