If {x_1}, {x_2}, {x_3}, ..., {x_n} are in A.P | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Given that ${x_1}, {x_2}, ..., {x_n}$ are in $A.P.$ with common difference $\alpha$,so ${x_{k+1}} - {x_k} = \alpha$ for all $k$.We can write $\sin

Vedclass · Accepted Answer

$\frac{\sin (n - 1)\alpha}{\cos {x_1}\cos {x_n}}$

Vedclass · Answer

(A) Given that ${x_1}, {x_2}, ..., {x_n}$ are in $A.P.$ with common difference $\alpha$,so ${x_{k+1}} - {x_k} = \alpha$ for all $k$.We can write $\sin \alpha = \sin({x_{k+1}} - {x_k})$.Thus,the expression becomes $\sum_{k=1}^{n-1} \sin \alpha \sec {x_k} \sec {x_{k+1}} = \sum_{k=1}^{n-1} \frac{\sin({x_{k+1}} - {x_k})}{\cos {x_k} \cos {x_{k+1}}}$.Using the identity $	an A - 	an B = \frac{\sin(A - B)}{\cos A \cos B}$,we get $\sum_{k=1}^{n-1} (	an {x_{k+1}} - 	an {x_k})$.This is a telescoping series: $(	an {x_2} - 	an {x_1}) + (	an {x_3} - 	an {x_2}) + ... + (	an {x_n} - 	an {x_{n-1}})$.After cancellation,we are left with $	an {x_n} - 	an {x_1} = \frac{\sin({x_n} - {x_1})}{\cos {x_n} \cos {x_1}}$.Since ${x_n} = {x_1} + (n-1)\alpha$,then ${x_n} - {x_1} = (n-1)\alpha$.Therefore,the value is $\frac{\sin(n - 1)\alpha}{\cos {x_1} \cos {x_n}}$.

If ${x_1}, {x_2}, {x_3}, ..., {x_n}$ are in $A.P.$ whose common difference is $\alpha$,then the value of $\sin \alpha (\sec {x_1}\sec {x_2} + \sec {x_2}\sec {x_3} + ... + \sec {x_{n - 1}}\sec {x_n}) = $

Similar Questions

$\sin (\beta + \gamma - \alpha ) + \sin (\gamma + \alpha - \beta ) + \sin (\alpha + \beta - \gamma ) - \sin (\alpha + \beta + \gamma ) = $

The value of $\frac{4 \sin 9^{\circ} \sin 21^{\circ} \sin 39^{\circ} \sin 51^{\circ} \sin 69^{\circ} \sin 81^{\circ}}{\sin 54^{\circ}}$ is equal to:

If $\tan \theta - \cot \theta = a$ and $\sin \theta + \cos \theta = b,$ then ${({b^2} - 1)^2}({a^2} + 4)$ is equal to

If $\sin x + \sin^2 x = 1$,then the value of $\cos^{12} x + 3\cos^{10} x + 3\cos^8 x + \cos^6 x - 2$ is equal to

The extreme values of $1 + 4 \sin \theta + 3 \cos \theta$ are:

Vedclass Test Series

Exam Paper Generator

Online Exam Module