MathDB
Problems
Contests
National and Regional Contests
India Contests
India LIMIT
2020 LIMIT
2020 LIMIT Category 2
18
18
Part of
2020 LIMIT Category 2
Problems
(1)
n \choose 1 sin (a) + n \choose 2 sin (2a) +...+ n \choose n sin (na)
Source: LIMIT 2020 Cat 2 Objective P18
6/9/2020
Evaluate the following sum:
(
n
1
)
n \choose 1
(
1
n
)
sin
(
a
)
+
\sin (a) +
sin
(
a
)
+
(
n
2
)
n \choose 2
(
2
n
)
sin
(
2
a
)
+
.
.
.
+
\sin (2a) +...+
sin
(
2
a
)
+
...
+
(
n
n
)
n \choose n
(
n
n
)
sin
(
n
a
)
\sin (na)
sin
(
na
)
(A)
2
n
cos
n
(
a
2
)
sin
(
n
a
2
)
2^n \cos^n \left(\frac{a}{2}\right)\sin \left(\frac{na}{2}\right)
2
n
cos
n
(
2
a
)
sin
(
2
na
)
(B)
2
n
sin
n
(
a
2
)
cos
(
n
a
2
)
2^n \sin^n \left(\frac{a}{2}\right)\cos \left(\frac{na}{2}\right)
2
n
sin
n
(
2
a
)
cos
(
2
na
)
(C)
2
n
sin
n
(
a
2
)
sin
(
n
a
2
)
2^n \sin^n \left(\frac{a}{2}\right)\sin \left(\frac{na}{2}\right)
2
n
sin
n
(
2
a
)
sin
(
2
na
)
(D)
2
n
cos
n
(
a
2
)
cos
(
n
a
2
)
2^n \cos^n \left(\frac{a}{2}\right)\cos \left(\frac{na}{2}\right)
2
n
cos
n
(
2
a
)
cos
(
2
na
)
trigonometry
Sum
limit
calculus