Question

pls answer me friends​

Answer 1

Given Question :-

Evaluate :-

$\rm :\longmapsto\:1 + cos10 + cos20 + cos30$

$\green{\large\underline{\sf{Solution-}}}$

Given Trigonometric expression is

$\rm :\longmapsto\:1 + cos10 + cos20 + cos30$

can be re-arranged as

$\rm \:  =  \: (1 + cos30) + (cos20 + cos10)$

We know,

$\boxed{ \tt{ \: 1 + cos2x = {2cos}^{2}x \: }} \\ \\ and \\ \\ \boxed{ \tt{ \: cosx + cosy = 2cos\bigg[\dfrac{x + y}{2} \bigg]cos\bigg[\dfrac{x - y}{2} \bigg]}}$

So, using this, we get

$\rm \:  =  \: 2 {cos}^{2}\dfrac{30}{2} + 2cos\bigg[\dfrac{20 + 10}{2} \bigg]cos\bigg[\dfrac{20 - 10}{2} \bigg]$

$\rm \:  =  \: 2 {cos}^{2}15 + 2cos\bigg[\dfrac{30}{2} \bigg]cos\bigg[\dfrac{10}{2} \bigg]$

$\rm \:  =  \: 2 {cos}^{2}15 + 2cos15 \: cos5$

$\rm \:  =  \: 2 {cos}^{2}15 + 2cos15 \: cos5$

$\rm \:  =  \: 2 \: cos15 \: [cos15 + cos5]$

$\rm \:  =  \: 2 \: cos15 \: \bigg[2 \: cos\dfrac{15 + 5}{2} \: cos\dfrac{15 - 5}{2} \bigg]$

$\rm \:  =  \: 4 \: cos15 \: \bigg[ \: cos\dfrac{20}{2} \: cos\dfrac{10}{2} \bigg]$

$\rm \:  =  \: 4 \: cos15 \: cos10 \: cos5$

$\rm \:  =  \: 4 \: cos5 \: cos10 \: cos15$

Hence,

$\boxed{ \tt{1 + cos10 + cos20 + cos30 =4cos5 \: cos10 \: cos15}}$

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Additional Information :-

$\boxed{ \tt{ \: sinx + siny = 2sin\bigg[\dfrac{x + y}{2} \bigg]cos\bigg[\dfrac{x - y}{2} \bigg]}}$

$\boxed{ \tt{ \: sinx - siny = 2cos\bigg[\dfrac{x + y}{2} \bigg]sin\bigg[\dfrac{x - y}{2} \bigg]}}$

$\boxed{ \tt{ \: cosx - cosy = - \: 2sin\bigg[\dfrac{x + y}{2} \bigg]sin\bigg[\dfrac{x - y}{2} \bigg]}}$