Sum of the series -r-188 -1 r-11-sin 2 r-1 -sin 21- is equal to

The correct option is A 2^∘/sin 2^∘ ∑_88^r=1( 1)^r+11/sin ^2(r+1)^∘ sin ^21^∘ = (1/sin 1^∘sin 3^∘+1/sin 3^∘sin 5^∘+...+1/sin 87^∘sin 89^∘ nbsp ...

158

You visited us 158 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Summation by Sigma Method

Sum of the se...

Question

Sum of the series $88 \sum r = 1 {(- 1)}^{r + 1} \frac{1}{{sin}^{2} {(r + 1)}^{\circ} - {sin}^{2} 1^{\circ}}$ is equal to

\frac{cot 2^{\circ}}{sin 2^{\circ}}

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

\frac{- cot 2^{\circ}}{sin 2^{\circ}}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

cot 2^{\circ}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

\frac{cot 2^{\circ}}{{sin}^{2} 2^{\circ}}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is A $\frac{cot 2^{\circ}}{sin 2^{\circ}}$
$r = 1 \sum 88 (- 1)^{r + 1} \frac{1}{{sin}^{2} (r + 1)^{\circ} - {sin}^{2} 1^{\circ}}$
$= (\frac{1}{sin 1^{\circ} sin 3^{\circ}} + \frac{1}{sin 3^{\circ} sin 5^{\circ}} + . . . + \frac{1}{sin 87^{\circ} sin 89^{\circ}}) - (\frac{1}{sin 2^{\circ} sin 4^{\circ}} + \frac{1}{sin 4^{\circ} sin 6^{\circ}} + . . . . . + \frac{1}{sin 88^{\circ} sin 90^{\circ}})$
$= \frac{1}{sin 2^{\circ}} [(cot 1^{\circ} - cot 3^{\circ}) + (cot 3^{\circ} - cot 5^{\circ}) + . . . . + (cot 87^{\circ} - cot 89^{\circ})]$
$= [(cot 2^{\circ} - cot 4^{\circ}) + (cot 4^{\circ} - cot 6^{\circ}) + . . . . . . . . + (cot 88^{\circ} - cot 90^{\circ})]$
$= \frac{cot 2^{\circ} - 0}{sin 2^{\circ}} = \frac{cot 2^{\circ}}{sin 2^{\circ}}$

Suggest Corrections

Similar questions

(\frac{1}{{sec}^{2} α - {cos}^{2} α} + \frac{1}{csc^{2} α - {sin}^{2} α}) \times {sin}^{2} α {cos}^{2} α = \frac{1 - {sin}^{2} α {cos}^{2} α}{2 + {sin}^{2} α {cos}^{2} α}

Which of the following best describes the terms of the series? $\frac{1}{2}$ $^{n} C_{1}$ - $\frac{2}{3}$ $^{n} C_{2}$ + $\frac{3}{4}$ $^{n} C_{3}$ ................... $(- 1)^{n + 1}$ $\frac{1}{n + 1}$ $^{n} C_{n}$

Q. The sum of the series

\sum_{r = 1}^{n} (- 1)^{r - 1} \cdot^{n} C_{r} (a - r)

is equal to

Q. If

sin θ + csc θ = 2,

then

{sin}^{2} θ + {csc}^{2} θ

is equal to

2

Q. Prove that:

{sin}^{2} 5^{o} + {sin}^{2} 10^{o} + {sin}^{2} 15^{o} . . . . + {sin}^{2} 85^{o} + {sin}^{2} 90^{o} = 9 \frac{1}{2}

Join BYJU'S Learning Program

Sum of the series 88∑r=1(−1)r+11sin2(r+1)∘−sin21∘ is equal to

Sum of the series $88 \sum r = 1 {(- 1)}^{r + 1} \frac{1}{{sin}^{2} {(r + 1)}^{\circ} - {sin}^{2} 1^{\circ}}$ is equal to