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Byju's Answer
Standard XII
Mathematics
Trigonometric Ratios of Compound Angles
If tan p x -t...
Question
If
tan
p
x
-
tan
q
x
=
0
, then the values of θ form a series in
(a) AP
(b) GP
(c) HP
(d) none of these
Open in App
Solution
(a) AP
Given:
tan
p
x
-
tan
q
x
=
0
⇒
tan
p
x
=
tan
q
x
⇒
sin
p
x
cos
p
x
=
sin
q
x
cos
q
x
⇒
sin
p
x
cos
q
x
=
sin
q
x
cos
p
x
⇒
1
2
sin
p
+
q
2
x
+
sin
p
-
q
2
x
=
1
2
sin
q
+
p
2
x
+
sin
q
-
p
2
x
Now,
sin
A
cos
B
=
1
2
sin
A
+
B
2
+
sin
A
-
B
2
⇒
sin
p
-
q
2
x
=
sin
q
-
p
2
x
⇒
sin
p
-
q
2
x
=
-
sin
p
-
q
2
x
⇒
2
sin
p
-
q
2
x
=
0
⇒
sin
p
-
q
2
x
=
0
⇒
p
-
q
2
x
=
n
π
,
n
∈
Z
⇒
x
=
2
nπ
(
p
-
q
)
,
n
∈
Z
Now, on putting the value of
n
, we get:
n
=
1
,
x
=
2
π
(
p
-
q
)
= a
1
n
=
2
,
x
=
4
π
(
p
-
q
)
= a
2
n
=
3
,
x
=
6
π
(
p
-
q
)
= a
3
n
=
4
,
x
=
8
π
(
p
-
q
)
= a
4
And so on.
Also,
d
=
a
2
-
a
1
=
4
π
(
p
-
q
)
-
2
π
(
p
-
q
)
=
2
π
(
p
-
q
)
d
=
a
3
-
a
2
=
6
π
(
p
-
q
)
-
4
π
(
p
-
q
)
=
2
π
(
p
-
q
)
d
=
a
4
-
a
3
=
8
π
(
p
-
q
)
-
6
π
(
p
-
q
)
=
2
π
(
p
-
q
)
And so on.
Thus,
x
forms a series in AP.
Suggest Corrections
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