Question

Match the column

$\begin{array}{cc}\text{Column A}& \text{Column B}\\ \text{1.cos}\frac{2\pi }{7}\text{+ cos}\frac{4\pi }{7}\text{+ cos}\frac{6\pi }{7}& \text{A.0}\\ \text{2.cos}\frac{\pi }{7}\text{+ cos}\frac{2\pi }{7}\text{+cos}\frac{3\pi }{7}\text{+ cos}\frac{4\pi }{7}\text{+ cos}\frac{5\pi }{7}\text{+ cos}\frac{6\pi }{7}& \text{B.}\frac{1}{2}\\ \text{3.sin}\frac{\pi }{11}\text{+ sin}\frac{3\pi }{11}\text{+ sin}\frac{5\pi }{11}\text{+ sin}\frac{7\pi }{11}\text{+ sin}\frac{9\pi }{11}& \text{C.}\frac{-1}{2}\end{array}$

• A. 1 - A, 2 - B, 3 - C
• B. 1 - C, 2 - A, 3 - B
• C. 1 - A, 2 - C, 3 - B
• D. 1 - B, 2 - A, 3 - C

Answer 1

The correct option is B

1 - C, 2 - A, 3 - B

We know, cos$\alpha$ + cos($\alpha +\beta$) + cos($\alpha +2\beta$) + .......... cos($\alpha +(n-1)\beta$)

just apply that formula

$\alpha$ = $\frac{2\pi }{7}$

$\beta$ = $\frac{2\pi }{7}$

n = 3

= $\frac{sin\left(\frac{3\times 2\pi }{2\times 7}\right)}{sin\left(\frac{2\pi }{2\times 7}\right)}\times cos(\frac{2\pi }{7}+\frac{(3-1)}{2}\times \frac{2\pi }{7})$

= $\frac{sin\left(\frac{3\pi }{7}\right)\times cos\left(\frac{4\pi }{7}\right)}{sin\left(\frac{\pi }{7}\right)}$

= $\frac{sin\frac{3\pi }{7}.cos(\pi -\frac{3\pi }{7})}{sin\left(\frac{\pi }{7}\right)}$ [cos $(\pi -\theta )$ = - cos $\theta$]

= $\frac{sin\frac{3\pi }{7}[-cos\frac{3\pi }{7}]}{sin\frac{\pi }{7}}$

= $\frac{-sin\frac{3\pi }{7}.cos\frac{3\pi }{7}}{sin\frac{\pi }{7}}$

We need to further simplify the expression, we observe that numerator is in the form of sin $\theta$ $\times$ cos$\theta$.Apply the formula sin2$\theta$ = 2sin$\theta$ $\times$ cos$\theta$

Multiplying 2 in numerator and denominator

We observe that we can have same trigonometric function in numerator and denominator by writing numerator as sin $(\pi -\theta )$ form because sin $(\pi -\theta )$ = sin $\theta$

we Know,

$\frac{sin\frac{6\pi }{14}\times 0}{sin\frac{\pi }{14}}$ = 0

3. cos$\frac{\pi }{11}$ + cos$\frac{3\pi }{11}$ + cos$\frac{5\pi }{11}$ + cos$\frac{7\pi }{11}$ + cos$\frac{9\pi }{11}$

= $\frac{\pi }{11}$, $\beta$ = $\frac{2\pi }{11}$,n=5

= $\frac{sin\left(\frac{5\times 2\pi }{2\times 11}\right)}{sin\left(\frac{2\pi }{2\times 11}\right)}.cos(\frac{\pi }{11}+\frac{4\times 2\pi }{2\times 11})$

Multiply and divide by 2 in the above expression.