Express cos 4 x cos 20 into the sum or difference,
Answers
Answer:
Convert sin 7α + sin 5α as a product.
Solution:
sin 7α + sin 5α
= 2 sin (7α + 5α)/2 cos (7α - 5α)/2, [Since, sin α + sin β = 2 sin (α + β)/2 cos (α - β)/2]
= 2 sin 6α cos α
2. Express sin 7A + sin 4A as a product.
Solution:
sin 7A + sin 4A
= 2 sin (7A + 4A)/2 cos (7A - 4A)/2
= 2 sin (11A/2) cos (3A)/2
Step-by-step explanation:
. Express the sum or difference as a product: cos ∅ - cos 3∅.
Solution:
cos ∅ - cos 3∅
= 2 sin (∅ + 3∅)/2 sin (3∅ - ∅)/2
= 2 sin 2∅ ∙ sin ∅.
4. Express cos 5θ - cos 11θ as a product.
Solution:
cos 5θ - cos 11θ
= 2 sin (5θ + 11θ)/2 sin (11θ - 5θ), [Since, cos α - cos β = 2 sin (α + β)/2 sin (β - α)/2]
= 2 sin 8θ sin 3θ
5. Prove that, sin 55° - cos 55° = √2 sin 10°
Solution:
L.H.S. = sin 55° - cos 55°
= sin 55° - cos (90° - 35°)
= sin 55° - sin 35°
= 2cos (55° + 35°)/2 sin (55° - 35°)/2
= 2 cos 45° sin 10°
= 2 ∙ 1/(√2) sin 10°
= √2 sin 10° = R.H.S. Proved
6. Prove that sin x + sin 3x + sin 5x + sin 7x = 4 cos x cos 2x sin 4x
Solution:
L.H.S. = sin x + sin 3x + sin 5x + sin 7x
= (sin 7x + sin x) + (sin 5x + sin 3x)
= 2 sin (7x + x)/2 cos (7x - x)/2 + 2 sin (5x + 3x)/2 cos (5x - 3x)/2
= 2 sin 4x cos 3x + 2 sin 4x cos x
= 2 sin 4x (cos 3x + cos x)
= 2 sin 4x ∙ 2 cos (3x + x)/2 cos (3x - x)/2
= 4 sin 4x cos 2x cos x = R.H.S.
7. Prove that, sin 20° + sin 140° - cos 10° = 0
Solution:
L.H.S. = sin 20° + sin 140° - cos 10°
= 2 ∙ sin (140° + 20°)/2 cos (140° - 20°)/2 - cos 10°, [Since sin C + sin D = 2 sin (C + D)/2 cos (C - D)/2]
= 2 sin 80° ∙ cos 60° - cos 10°
= 2 ∙ sin (90° - 10°) ∙ 1/2 - cos 10° [Since, cos 60° = 1/2]
= cos 10° - cos 10°
= 0 = R.H.S. Proved