Question

41 2 cos 2" 6+1 1+1 = (2 cos 0 – 1) (2 cos 20 – 1) (2 cos 220 - 1) 2 cos 0 +1 ... (2 cos 21-1 0-1)

JEE/Engineering Exams

Maths

Solution

4.0 (1 ratings)

( (2 cos theta-1)(2 cos 2 theta-1)left(2 cos 2^{2} theta-1right) cdotsleft(2 cos 2^{n-1} theta-1right) ) Muttiply and divide hy ( (2 cos theta+1) ) ( Rightarrow frac{(2 cos theta+1)(2 cos theta-1)(2 cos 2 theta-1) cdots-left(2 cos 2^{n-1} theta-1right)}{(2 cos theta+1)} ) ( Rightarrow frac{left(4 cos ^{2} theta-1right)(2 cos 2 theta-1)-cdotsleft(2 cos 2^{n-1} theta-1right)}{(2 cos theta+1)} ) ( Rightarrow ) Replace ( 4 cos ^{2} theta=2(1+cos 2 theta) ) ( Rightarrow frac{(2 cos 2 theta+1)(2 cos 2 theta-1)}{(2 cos theta+1)} ) ( Rightarrow quad frac{left(4 cos ^{2} 20 piright)left(2 cos 2^{2} 0-1right)}{(2 cos theta+1)} cdotleft(because^{prime}-left(4 cos ^{2} theta-1right)=2 cos 2^{2} theta+1right. ) Similonly doing this, we com that this gres ( 10 Rightarrowleft(2 cos _{2} n_{0}+1right) / 2 cos theta+1=14 )