Let f: R→R be defined as f(x+y)+f(x−y)= 2f(x)f(y), f(12)=−1. Then, the value of ∑k=1201sin(k)sin(k+f(k))is equal to:

# $\begin{array}{l}Let\text{\hspace{0.17em}}f:\text{\hspace{0.17em}}R\to R\text{\hspace{0.17em}}be\text{\hspace{0.17em}}defined\text{\hspace{0.17em}}as\text{\hspace{0.17em}}f\left(x+y\right)+f\left(x-y\right)=\text{\hspace{0.17em}}2f\left(x\right)f\left(y\right),\text{\hspace{0.17em}}f\left(\frac{1}{2}\right)=-1.\text{\hspace{0.17em}}Then,\text{\hspace{0.17em}}the\text{\hspace{0.17em}}value\text{\hspace{0.17em}}of\text{\hspace{0.17em}}\sum _{k=1}^{20}\frac{1}{\mathrm{sin}\left(k\right)\mathrm{sin}\left(k+f\left(k\right)\right)}\\ is\text{\hspace{0.17em}}equal\text{\hspace{0.17em}}to:\end{array}$

1. A

${\mathrm{sec}}^{2}\left(21\right)\mathrm{sin}\left(20\right)\mathrm{sin}\left(2\right)$

2. B

$\mathrm{cos}e{c}^{2}\left(1\right)\mathrm{cos}ec\left(21\right)\mathrm{sin}\left(20\right)$

3. C

${\mathrm{sec}}^{2}\left(1\right)\mathrm{sec}\left(21\right)\mathrm{cos}\left(20\right)$

4. D

$\mathrm{cos}e{c}^{2}\left(21\right)\mathrm{cos}\left(20\right)\mathrm{cos}\left(2\right)$

Fill Out the Form for Expert Academic Guidance!l

+91

Live ClassesBooksTest SeriesSelf Learning

Verify OTP Code (required)

## Related content

 Matrices and Determinants_mathematics Critical Points Solved Examples Type of relations_mathematics

+91

Live ClassesBooksTest SeriesSelf Learning

Verify OTP Code (required)