Page No 8.17:
Question 1:
Express each of the following as the product of sines and cosines:
(i) sin 12x + sin 4x
(ii) sin 5x − sin x
(iii) cos 12x + cos 8x
(iv) cos 12x − cos 4x
(v) sin 2x + cos 4x
Answer:
(i)
(ii)
(iii)
(iv)
(v)
Page No 8.17:
Question 2:
Prove that:
(i) sin 38° + sin 22° = sin 82°
(ii) cos 100° + cos 20° = cos 40°
(iii) sin 50° + sin 10° = cos 20°
(iv) sin 23° + sin 37° = cos 7°
(v) sin 105° + cos 105° = cos 45°
(vi) sin 40° + sin 20° = cos 10°
Answer:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Page No 8.17:
Question 3:
Prove that:
(i) cos 55° + cos 65° + cos 175° = 0
(ii) sin 50° − sin 70° + sin 10° = 0
(iii) cos 80° + cos 40° − cos 20° = 0
(iv) cos 20° + cos 100° + cos 140° = 0
(v)
(vi)
(vii) sin 80° − cos 70° = cos 50°
(viii) sin 51° + cos 81° = cos 21°
Answer:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
Page No 8.18:
Question 4:
Prove that:
(i)
(ii)
Answer:
(i)
(ii)
Page No 8.18:
Question 5:
Prove that:
(i)
(ii) sin 47° + cos 77° = cos 17°
Answer:
(i)
(ii)
Page No 8.18:
Question 6:
Prove that:
(i) cos 3A + cos 5A + cos 7A + cos 15A = 4 cos 4A cos 5A cos 6A
(ii) cos A + cos 3A + cos 5A + cos 7A = 4 cos A cos 2A cos 4A
(iii) sin A + sin 2A + sin 4A + sin 5A = 4 cos cos sin 3A
(iv) sin 3A + sin 2A − sin A = 4 sin A cos cos
(v) cos 20° cos 100° + cos 100° cos 140° − 140° cos 200° = −
(vi)
(vii)
Answer:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Hence, LHS = RHS
(vii)
Hence, LHS = RHS
Disclaimer: The given question is incorrect. The correct question should be .
Page No 8.18:
Question 7:
Prove that:
(i)
(ii)
(iii)
(iv)
(v)
Answer:
(i)
(ii)
(iii)
(iv)
(v)
Page No 8.18:
Question 8:
Prove that:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
(xi)
Answer:
Page No 8.19:
Question 9:
Prove that:
(i)
(ii) cos (A + B + C) + cos (A − B + C) + cos (A + B − C) + cos (− A + B + C) = 4 cos A cos B cos C
Answer:
Page No 8.19:
Question 10:
Answer:
Given:
sin A + sin B = .....(i)
cos A + cos B = .....(ii)
Dividing (i) by (ii):
Page No 8.19:
Question 11:
If cosec A + sec A = cosec B + sec B, prove that tan A tan B = .
Answer:
Given:
Page No 8.19:
Question 12:
.
Answer:
Given:
sin 2A = λ sin 2B
Hence proved.
Page No 8.19:
Question 13:
Prove that:
(i)
(ii) sin (B − C) cos (A − D) + sin (C − A) cos (B − D) + sin (A − B) cos (C − D) = 0
Answer:
Page No 8.19:
Question 14:
Answer:
Page No 8.19:
Question 15:
If cos (α + β) sin (γ + δ) = cos (α − β) sin (γ − δ), prove that cot α cot β cot γ = cot δ
Answer:
Page No 8.19:
Question 16:
If y sin Ï = x sin (2θ + Ï), prove that (x + y) cot (θ + Ï) = (y − x) cot θ.
Answer:
Given:
y sin Ï = x sin (2θ + Ï)
Page No 8.19:
Question 17:
If cos (A + B) sin (C − D) = cos (A − B) sin (C + D), prove that tan A tan B tan C + tan D = 0.
Answer:
cos (A + B) sin (C − D) = cos (A − B) sin (C + D)
[cosA cosB − sinA sinB] [sinC cosD − cosC sinD] = [cosA cosB + sinA sinB] [sinC cosD + cosC sinD]
Page No 8.19:
Question 18:
If , prove that . [NCERT EXEMPLAR]
Answer:
Page No 8.19:
Question 19:
If , prove that . [NCERT EXEMPLAR]
Answer:
Given:
Applying componendo and dividendo, we get
Page No 8.20:
Question 1:
cos 40° + cos 80° + cos 160° + cos 240° =
(a) 0
(b) 1
(c)
(d)
Answer:
(d)
Page No 8.20:
Question 2:
sin 163° cos 347° + sin 73° sin 167° =
(a) 0
(b)
(c) 1
(d) None of these
Answer:
(b)
Page No 8.20:
Question 3:
If sin 2 θ + sin 2 Ï = and cos 2 θ + cos 2 Ï = , then cos2 (θ − Ï) =
(a)
(b)
(c)
(d)
Answer:
(b)
Given:
sin 2θ + sin 2Ï = .....(i)
and
cos 2θ + cos 2Ï = .....(ii)
Squaring and adding (i) and (ii), we get:
(sin 2θ + sin 2Ï)2 + (cos 2θ + cos 2Ï)2 =
Page No 8.20:
Question 4:
The value of cos 52° + cos 68° + cos 172° is
(a) 0
(b) 1
(c) 2
(d) 3/2
Answer:
(a) 0
Page No 8.20:
Question 5:
The value of sin 78° − sin 66° − sin 42° + sin 60° is
(a)
(b)
(c) −1
(d) None of these
Answer:
(d) None of these
Page No 8.20:
Question 6:
If sin α + sin β = a and cos α − cos β = b, then tan =
(a)
(b)
(c)
(d) None of these
Answer:
(b)
Given:
sin α + sin β = a .....(i)
cos α − cos β = b .....(ii)
Dividing (i) by (ii):
Page No 8.21:
Question 7:
cos 35° + cos 85° + cos 155° =
(a) 0
(b)
(c)
(d) cos 275°
Answer:
(a) 0
Page No 8.21:
Question 8:
The value of sin 50° − sin 70° + sin 10° is equal to
(a) 1
(b) 0
(c) 1/2
(d) 2
Answer:
(b) 0
Page No 8.21:
Question 9:
sin 47° + sin 61° − sin 11° − sin 25° is equal to
(a) sin 36°
(b) cos 36°
(c) sin 7°
(d) cos 7°
Answer:
(d) cos 7°
Page No 8.21:
Question 10:
If cos A = m cos B, then =
(a)
(b)
(c)
(d) None of these
Answer:
(c)
Page No 8.21:
Question 11:
If A, B, C are in A.P., then =
(a) tan B
(b) cot B
(c) tan 2 B
(d) None of these
Answer:
(b) cot B
Since A,B and C are in A.P,
Page No 8.21:
Question 12:
If sin (B + C − A), sin (C + A − B), sin (A + B − C) are in A.P., then cot A, cot B and cot C are in
(a) GP
(b) HP
(c) AP
(d) None of these
Answer:
(b) HP
Given:
sin (B + C − A), sin (C + A − B) and sin (A + B − C) are in A.P.
Hence, cotA, cotB and cotC are in HP.
Page No 8.21:
Question 13:
If sin x + sin y = (cos y − cos x), then sin 3x + sin 3y =
(a) 2 sin 3x
(b) 0
(c) 1
(d) none of these
Answer:
We have,
sin x + sin y = (cos y − cos x)
Page No 8.21:
Question 14:
The value of is given by
(a)
(b) 1
(c)
(d)
Answer:
â
Hence, the correct answer is option A.
Page No 8.21:
Question 1:
The value of is _________.
Answer:
â
Page No 8.21:
Question 2:
If tan (A + B) = p, tan (A – B) = q, then the value of tan 2A in terms of p and q is ___________.
Answer:
Given (A + B) = p
tan (A – B) = q
Page No 8.21:
Question 3:
If 1 + cos 2x + cos 4x + cos 6x = k cos x cos 2x cos 3x, then k = ____________.
Answer:
Given 1 + cos 2x + cos 4x + cos 6x = k cos x cos 2x cos 3x
Consider,
Page No 8.21:
Question 4:
The value of sin 50° – sin 70° + sin 10° is ___________.
Answer:
sin 50° – sin 70° + sin 10°
Page No 8.21:
Question 5:
The value of is ________________.
Answer:
â
Using identity:- 2 cos x cos y = cos (x + y) + cos (x – y)
Hence, the answer is 0.
Page No 8.22:
Question 1:
If (cos α + cos β)2 + (sin α + sin β)2 = , write the value of λ.
Answer:
(cos α + cos β)2 + (sin α + sin β)2 =
Consider LHS:
(cos α + cos β)2 + (sin α + sin β)2
Page No 8.22:
Question 2:
Write the value of sin sin .
Answer:
sin sin
Page No 8.22:
Question 3:
If sin A + sin B = α and cos A + cos B = β, then write the value of tan .
Answer:
Given:
sin A + sin B = α .....(i)
cos A + cos B = β .....(ii)
Dividing (i) by (ii):
Page No 8.22:
Question 4:
If cos A = m cos B, then write the value of .
Answer:
Page No 8.22:
Question 5:
Write the value of the expression .
Answer:
Page No 8.22:
Question 6:
If A + B = and cos A + cos B = 1, then find the value of cos .
Answer:
Given:
A + B =
and cos A + cos B = 1
Page No 8.22:
Question 7:
Write the value of
Answer:
Page No 8.22:
Question 8:
If sin 2A = λ sin 2B, then write the value of .
Answer:
Given:
sin 2A = λ sin 2B
Page No 8.22:
Question 9:
Write the value of .
Answer:
Page No 8.22:
Question 10:
If cos (A + B) sin (C − D) = cos (A − B) sin (C + D), then write the value of tan A tan B tan C.
Answer:
cos (A + B) sin (C − D) = cos (A − B) sin (C + D)
[cosA cosB − sinA sinB] [sinC cosD − cosC sinD] = [cosA cosB + sinA sinB] [sinC cosD + cosC sinD]
Page No 8.6:
Question 1:
Express each of the following as the sum or difference of sines and cosines:
(i) 2 sin 3x cos x
(ii) 2 cos 3x sin 2x
(iii) 2 sin 4x sin 3x
(iv) 2 cos 7x cos 3x
Answer:
(i)
(ii)
(iii)
(iv)
Page No 8.6:
Question 2:
Prove that:
(i)
(ii)
(iii)
Answer:
(i)
(ii)
(iii)
Page No 8.6:
Question 3:
Show that :
(i)
(ii)
Answer:
(i)
(ii)
Page No 8.7:
Question 4:
Answer:
Page No 8.7:
Question 5:
Prove that:
(i) cos 10° cos 30° cos 50° cos 70° =
(ii) cos 40° cos 80° cos 160° =
(iii) sin 20° sin 40° sin 80° =
(iv) cos 20° cos 40° cos 80° =
(v) tan 20° tan 40° tan 60° tan 80° = 3
(vi) tan 20° tan 30° tan 40° tan 80° = 1
(vii) sin 10° sin 50° sin 60° sin 70° =
(viii) sin 20° sin 40° sin 60° sin 80° =
Answer:
(i)
(ii)
(iii)
(iv)
(v)
LHS = tan 20° tan 40° tan 60° tan 80°
(vi)
LHS = tan 20° tan 30° tan 60° tan 80°
(vii)
(viii)
Page No 8.7:
Question 6:
Show that:
(i) sin A sin (B − C) + sin B sin (C − A) + sin C sin (A − B) = 0
(ii) sin (B − C) cos (A − D) + sin (C − A) cos (B − D) + sin (A − B) cos (C − D) = 0
Answer:
(i)
(ii)
Page No 8.7:
Question 7:
Prove that
Answer:
Page No 8.7:
Question 8:
If α + β = , show that the maximum value of cos α cos β is .
Answer:
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