Page 43 - mathsvol1ch1to3ans

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(ii) sin(30 + θ) + cos(60 + θ) = cos θ.
Solution:
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sin(45 + θ) − sin(45 − θ) = sin(45 ) cos θ + sin θ cos(45 ) − sin(45 ) cos θ + sin θ cos(45 )
2 sin θ √
= √ = 2 sin θ
2
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sin(30 + θ) + cos(60 + θ) = sin(30 ) cos θ + sin θ cos(30 ) + cos(60 ) cos θ − sin(60 ) sin θ
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= sin(30 ) cos θ + sin θ cos(30 ) + sin(30 ) cos θ − cos(30 ) sin θ
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= 2 sin(30 ) cos θ
= cos θ
10. If a cos(x + y) = b cos(x − y), show that (a + b) tan x = (a − b) cot y.
Solution:
a cos(x + y) = b cos(x − y)
a cos(x) cos(y) − a sin(x) sin(y) = b cos(x) cos(y) + b sin(x) sin(y)
(a − b) cos(x) cos(y) = (a + b) sin(x) sin(y)
(a − b) = (a + b) tan(x) tan(y)

(a − b) cot(y) = (a + b) tan(x)
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11. Prove that sin 105 + cos 105 = cos 45 .
Solution:
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sin(105) + cos(105) = sin(45) cos(60) + sin(60) cos(45) + cos(45) cos(60) − sin(60) sin(45)
√ √
1 3 1 3
= √ + √ + √ − √
2 2 2 2 2 2 2 2
1
= √
2
= cos 45 ◦
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12. Prove that sin 75 − sin 15 = cos 105 + cos 15 .
Solution:
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sin 75 − sin 15 = sin(90 − 15 ) − {− cos(90 + 15 )} = cos 15 + cos 105 ◦
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13. Show that tan 75 + cot 75 = 4.
Solution: