Prove that:
4. (cos x – cos y)
2
+ (sin x – sin y)
2
= 4 sin2
2
x y −
5. sin x + sin 3x + sin 5x + sin 7x = 4 cos x cos 2x sin 4x
6 . (sin7 sin5 ) (sin 9 sin3 ) tan 6
(cos7 cos5 ) (cos9 cos3 )
x x x x
x
Prove that:
1. 0
13
5π
cos
13
3π
cos
13
9π
cos
13
π
cos2 =++
2. (sin 3x + sin x) sin x + (cos 3x – cos x) cos x = 0
3. (cos x + cos y)
2
+ (sin x – sin y)
2
= 4 cos2
2
x y
Prove that cos2 x + cos2 π 2 π 3
cos
3 3 2
x x
+ + − =
If
3 3π
tan = , π < <
4 2
x x , find the value of sin
x
2
, cos
x
2
and tan
x
2
If sin x =
3
5
, cos y = −
12
13
, where x and y both lie in second quadrant,
find the value of sin (x + y).
Prove the following:-
22. cot x cot 2x – cot 2x cot 3x – cot 3x cot x = 1
23.
2
2 4
4tan (1 tan ) tan 4
1 6 tan tan
x x
x
x x
−
=
− +
24. cos 4x = 1 – 8sin2
x cos2
x
25. cos 6x = 32 cos6
x – 48cos4
x + 18 cos2
x – 1
Prove the following:-
16.
cos cos
sin sin
sin
cos
9 5
17 3
2
10
x x
x x
x
x
−
−
= − 17.
sin sin
cos cos
tan
5 3
5 3
4
x x
x x
x
+
+
=
18.
sin sin
cos cos
tan
x y
x y
− x y
+
=
−
2
19.
sin sin
cos cos
tan
x x
x x
x
+
+
=
3
3
2
20.
sin sin
sin cos
sin
x x
x x
x
−
−
=
3
2 2 2 21.
cos cos cos
sin sin sin
cot
4 3 2
4 3 2
3
x x x
x x x
x
Prove the following:
10. sin (n + 1)x sin (n + 2)x + cos (n + 1)x cos (n + 2)x = cos x
11.
3 3
cos cos 2 sin
4 4
x x x
π π
+ − − = −
12. sin2
6x – sin2 4x = sin 2x sin 10x 13. cos2
2x – cos2
6x = sin 4x sin 8x
14. sin2 x + 2 sin 4x + sin 6x = 4 cos2 x sin 4x
15. cot 4x (sin 5x + sin 3x) = cot x (sin 5x – sin 3x)
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