if a+b+c=180 then cosa+cosb-cosc=
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Answer:
As
A
,
B
and
C
are angles of a triangle, we have
A
+
B
+
C
=
180
and
A
+
B
=
180
−
C
or
A
+
B
2
=
90
−
C
2
Hence
cos
A
+
cos
B
+
cos
C
=
2
cos
(
A
+
B
2
)
cos
(
A
−
B
2
)
+
1
−
2
sin
2
(
C
2
)
−
1
=
2
cos
(
90
−
C
2
)
cos
(
A
−
B
2
)
+
1
−
2
sin
2
(
C
2
)
=
2
sin
(
C
2
)
cos
(
A
−
B
2
)
+
1
−
2
sin
2
(
C
2
)
=
2
sin
(
C
2
)
(
cos
(
A
−
B
2
)
−
sin
(
C
2
)
)
+
1
=
2
sin
(
C
2
)
(
cos
(
A
−
B
2
)
−
cos
(
A
+
B
2
)
)
+
1
=
2
sin
(
C
2
)
(
2
sin
(
A
2
)
sin
(
B
2
)
)
+
1
=
1
+
4
sin
(
A
2
)
sin
(
B
2
)
sin
(
C
2
)
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