12. Evaluate : - cos - cos 30 1 cos...
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12. Evaluate : - cos - cos 30 1 cosa - cos 50 - +.... + cos 0 -cos 70 cos - cos(2n +1) 13 volunt

JEE/Engineering Exams
Maths
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( begin{array}{l}begin{aligned} & sum_{n=1}^{n} frac{1}{cos theta-cos (2 n+1) theta} &=sum_{n=1}^{n} frac{1}{2 sin (2 n theta) sin (2 n theta+2 theta)}=-sum_{n=1}^{infty} frac{sin (2 n theta-(2 n theta+2 theta))}{2 sin 2 theta} &=sum_{n=1}^{n} frac{sin (2 n theta+2 theta-2 n theta)}{2 sin 2 theta sin (2 n theta) sin (2 n theta+2 theta)} &=frac{1}{2} sum_{n=1}^{n} frac{sin (2 n theta+2 theta) cos (2 n theta)-cos (2 n theta+2 theta) sin (2 n theta)}{sin (2 n theta) sin (2 n theta+2 theta)} &=frac{1}{2 sin 2 theta} sum_{n=1}^{n}(cot (2 n theta)-cot (2 n theta+2 theta)) =& frac{1}{2 sin 2 theta}left{begin{array}{c}(cot 2 theta-cot 4 theta)+(cot 4 theta-cot 6 theta)+(cot 6 theta-cot theta theta) +cdots+frac{cot (2 n theta)-cot (2 n theta+2 theta)}{sin 2 theta}(cot 2 theta-cot (2 n theta+2 theta)}end{array}right) =& frac{1}{2 sin 2 theta}left(frac{cos 2 theta sin (2 n theta+2 theta)-sin (2 theta) cos (2 n theta+2 theta)}{sin 2 theta sin (2 n theta+2 theta)}right.end{aligned} = & frac{sin (2 n theta)}{2 sin ^{2} 2 theta sin (2 n theta+2 theta)}end{array} )
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