Page 53 - mathsvol1ch1to3ans

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sin 2x + sin x + sin 3x = sin 2x + 2 sin 2x cos x = sin 2x (1 + 2 cos x)

sin 4x + sin 2x
8. Prove that = tan 3x.
cos 4x + cos 2x

Solution:
sin 4x + sin 2x 2 sin 3x cos x
= = tan 3x
cos 4x + cos 2x 2 cos 3x cos x

9. Prove that 1 + cos 2x + cos 4x + cos 6x = 4 cos x cos 2x cos 3x.
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Solution:
2
1 + cos 6x + cos 2x + cos 4x = 2 cos 3x + 2 cos 3x cos x
= 2 cos 3x (cos 3x + cos x)

= 2 cos 3x (2 cos 2x cos x)

= 4 cos x cos 2x cos 3x

θ 7θ 3θ 11θ
10. Prove that sin sin + sin sin = sin 2θ sin 5θ.
2 2 2 2
Solution:

θ 7θ 3θ 11θ
2 sin sin + 2 sin sin = (cos 3θ − cos 4θ) + (cos 4θ − cos 7θ)
2 2 2 2
= cos 3θ − cos 7θ

= 2 sin 5θ sin 2θ

θ 7θ 3θ 11θ
Hence sin sin + sin sin = sin 2θ sin 5θ
2 2 2 2
1
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11. Prove that cos (30 − A) cos (30 + A) + cos (45 − A) cos (45 + A) = cos 2A + .
4
Solution:
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2 cos (30 − A) cos (30 + A) + 2 cos (45 − A) cos (45 + A) = (cos 60 + cos 2A) + (cos 90 + cos 2A)
1
= + 2 cos 2A
2
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Hence cos (30 − A) cos (30 + A) + cos (45 − A) cos (45 + A) = + cos 2A
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sin x + sin 3x + sin 5x + sin 7x
12. Prove that = tan 4x.
cos x + cos 3x + cos 5x + cos 7x
Solution:
sin x + sin 7x + sin 3x + sin 5x 2 sin 4x cos 3x + 2 sin 4x cos x
=
cos x + cos 7x + cos 3x + cos 5x 2 cos 4x cos 3x + 2 cos 4x cos x
2 sin 4x (cos 3x + cos x)
=
2 cos 4x (cos 3x + cos x)
= tan 4x