Page No 216:
Question 33:
Answer:
Page No 216:
Question 34:
Prove each of the following identities:
Answer:
Page No 216:
Question 35:
Answer:
Page No 314:
Question 1:
(i) (1 − cos2θ) cosec2θ = 1
(ii) (1 + cot2θ) sin2θ = 1
Answer:
Page No 314:
Question 2:
(i) (sec2θ − 1) cot2θ = 1
(ii) (sec2θ − 1) (cosec2θ − 1) = 1
(iii) (1− cos2θ) sec2θ = tan2θ
Answer:
Page No 314:
Question 3:
(i)
(ii)
Answer:
Page No 314:
Question 4:
(i) (1 + cos θ) (1 − cos θ) (1 + cot2θ) = 1
(ii) cosec θ (1 + cos θ) (cosec θ − cot θ) = 1
Answer:
Page No 314:
Question 5:
(i)
(ii)
Answer:
Page No 314:
Question 6:
(i) sec θ (1 − sin θ) (sec θ + tan θ) = 1
(ii) sin θ(1 + tan θ) + cos θ(1 + cot θ) = (sec θ + cosec θ)
Answer:
Page No 314:
Question 7:
Answer:
Hence, LHS = RHS
Page No 314:
Question 8:
Answer:
Page No 314:
Question 9:
Answer:
Hence, L.H.S. = R.H.S.
Page No 314:
Question 10:
Answer:
Hence, LHS = RHS
Page No 314:
Question 11:
Answer:
Hence, LHS = RHS
Page No 314:
Question 12:
Answer:
Hence, L.H.S. = R.H.S.
Page No 314:
Question 13:
Answer:
Hence, LHS = RHS
Page No 315:
Question 14:
Answer:
Hence, LHS = RHS
Page No 315:
Question 15:
(i)
(ii)
(iii)
Answer:
Page No 315:
Question 16:
(i)
(ii)
Answer:
Page No 315:
Question 17:
Answer:
∴ LHS = RHS
Hence proved.
Page No 315:
Question 18:
(i)
(ii)
Answer:
Page No 315:
Question 19:
(i)
(ii)
Answer:
Page No 315:
Question 20:
(i)
(ii)
Answer:
Page No 315:
Question 21:
Answer:
Page No 315:
Question 22:
(i)
(ii)
Answer:
Page No 315:
Question 23:
Answer:
Hence, LHS= RHS
Page No 315:
Question 24:
Answer:
Hence, LHS = RHS
Page No 315:
Question 25:
Answer:
Hence, L.H.S. = R.H.S.
Page No 316:
Question 26:
(i)
(ii)
Answer:
Page No 316:
Question 27:
(i)
(ii)
Answer:
Page No 316:
Question 28:
Answer:
Hence, LHS = RHS
Page No 316:
Question 29:
Answer:
Page No 316:
Question 30:
Answer:
Hence, LHS = RHS
Page No 316:
Question 31:
Answer:
Hence, LHS = RHS
Page No 316:
Question 32:
Answer:
Hence, LHS = RHS
Page No 316:
Question 36:
Answer:
Hence, LHS = RHS
Page No 316:
Question 37:
Show that none of the following is an identity:
(i) cos2θ + cos θ = 1
(ii) sin2θ + sin θ = 2
(iii) tan2θ + sin θ = cos2θ
Answer:
Page No 317:
Question 38:
Show that each of the following is an identity:
(i)
(ii)
Answer:
Page No 321:
Question 1:
If a cos θ + b sin θ = m and a sin θ − b cos θ = n, prove that (m2 + n2) = (a2 + b2).
Answer:
Page No 321:
Question 2:
If x = a sec θ + b tan θ and y = a tan θ + b sec θ, prove that (x2 − y2) = (a2 − b2).
Answer:
Page No 322:
Question 3:
prove that
Answer:
Page No 322:
Question 4:
If (sec θ + tan θ) = m and (sec θ − tan θ) = n, show that mn = 1.
Answer:
Page No 322:
Question 5:
If (cosec θ + cot θ) = m and (cosec θ − cot θ) = n, show that mn = 1.
Answer:
Page No 322:
Question 6:
If x = a cos3θ and y = b sin3θ, prove that
Answer:
Page No 322:
Question 7:
If (tan θ + sin θ) = m and (tan θ − sin θ) = n, prove that (m2 − n2)2 = 16mn.
Answer:
Page No 322:
Question 8:
If (cot θ + tan θ) = m and (sec θ − cos θ) = n, prove that (m2n)2/3 − (mn2)2/3 = 1.
Answer:
Page No 322:
Question 9:
If (cot θ + tan θ) = m and (sec θ − cos θ) = n, prove that (m2n)2/3 − (mn2)2/3 = 1.
Answer:
Page No 322:
Question 10:
If (cos θ + sin θ) = 1, prove that (cos θ − sin θ) = ± 1.
Answer:
Page No 322:
Question 11:
If tan A = n tan B and sin A = m sin B, prove that cos2A =
Answer:
Page No 322:
Question 12:
If (cosec θ − sin θ) = a3 and (sec θ − cos θ) = b3, prove that
Answer:
Page No 322:
Question 13:
If a cos3θ + 3a sin2θ cos θ = m and a sin3θ + 3a sin θ cos2θ = n, prove that
(m + n)2/3 + (m − n)2/3 = 2a2/3.
Answer:
Page No 322:
Question 14:
If (2 sin θ + 3 cos θ) = 2, show that (3 sin θ − 2 cos θ) = ± 3.
Answer:
Page No 322:
Question 15:
If sin θ + cos θ = sin (90° − θ), show that cot θ = .
Answer:
Page No 322:
Question 16:
If cos θ + sin θ = sin θ, show that sin θ − cos θ = cos θ.
Answer:
Page No 322:
Question 17:
If tan θ = , show that .
Answer:
Hence, LHS = RHS
Page No 323:
Question 1:
If sin A + sin2 A = 1, then (cos2 A + cos4 A) = ?
(a)
(b) 1
(c) 2
(d) 3
Answer:
(b) 1
Page No 324:
Question 2:
If cos A + cos2 A = 1, then (sin2 A + sin4 A) = ?
(a) 1
(b) 2
(c) 3
(d) 4
Answer:
(a) 1
Page No 324:
Question 3:
(sec A + tan A) (1 − sin A) = ?
(a) sin A
(b) cos A
(c) sec A
(d) cosec A
Answer:
(b) cos A
Page No 324:
Question 4:
(1 + tan A + sec A)(1 + cot A − cosec A) = ?
(a) 1
(b) 0
(c) 2
(d) 4
Answer:
(c) 2
Page No 324:
Question 5:
(a) tan2 A
(b) cot2 A
(c) sec2 A
(d) cosec2 A
Answer:
(a) tan2 A
Page No 324:
Question 6:
(sin A + cos A)2 + (sin A − cos A)2 = ?
(a) 1
(b) 2
(c) 3
(d) 4
Answer:
(c) 2
Page No 324:
Question 7:
(sec4 A − sec2 A) = ?
(a) tan4 A − tan2 A
(b) tan2 A + tan4 A
(c) tan2 A − tan4 A
(d) None of these
Answer:
(b) tan2 A + tan4 A
Page No 324:
Question 8:
(cos4θ − sin4θ)=?
(a) 1 − 2 sin2θ
(b) 1 − 2 cos2θ
(c) 2 − sin2θ
(d) 2 − cos2θ
Answer:
a) 1 − 2 sin2θ
Page No 324:
Question 9:
(a)
(b)
(c)
(d) None of these
Answer:
(b)
Page No 324:
Question 10:
(a) 2 sin θ
(b) 2 cos θ
(c) 2 sec θ
(d) 2 cosec θ
Answer:
(d) 2 cosec θ
Page No 324:
Question 11:
(a) (cos θ + sin θ)
(b) (cos θ − sin θ)
(c) 0
(d) 2 tan θ
Answer:
(a) (cos θ + sin θ)
Page No 324:
Question 12:
(a) (sin2θ − cos2θ)
(b) (cos2θ − sin2θ)
(c) (cot2θ − tan2θ)
(d) (tan2θ − cot2θ)
Answer:
(b) (cos2θ − sin2θ)
Page No 325:
Question 13:
(a) (sec A + tan A)
(b) (sec A − tan A)
(c) sec A tan A
(d) None of these
Answer:
(a) (sec A + tan A)
Page No 325:
Question 14:
(a) (sec A + tan A)
(b) (sec A − tan A)
(c) sec A tan A
(d) None to these
Answer:
(b) (sec A − tan A)
Page No 325:
Question 15:
(a) (cosec A − cot A)
(b) (cosec A + cot A)
(c) cosec A cot A
(d) None of these
Answer:
(a) (cosec A − cot A)
Page No 325:
Question 16:
(a) (cosec A − cot A)
(b) (cosec A + cot A)
(c) cosec A cot A
(d) None of these
Answer:
(b) (cosec A + cot A)
Page No 325:
Question 17:
(a) (sec A − tan A)
(b) (sec A + tan A)
(c) sec A tan A
(d) None of these
Answer:
(a) (sec A − tan A)
Page No 325:
Question 18:
(sin4θ − cos4θ + 1) cosec2θ = ?
(a) 4
(b) 3
(c) 2
(d) 1
Answer:
(c) 2
Page No 325:
Question 19:
If x = a cos θ and y = b sin θ, then (b2x2 + a2y2) = ?
(a) a2 + b2
(b) a2b2
(c) ab
(d) a4b4
Answer:
(b) a2b2
Page No 325:
Question 20:
If x = a sec θ cos ϕ, y = b θ sin ϕ and z = c tan θ, then
(a)
(b)
(c)
(d)
Answer:
(a)
Page No 325:
Question 21:
(a) 2 sin θ
(b) 2 cos θ
(c) 2 cosec θ
(d) 2 sec θ
Answer:
(c) 2 cosec θ
Page No 326:
Question 22:
If (sin θ + cos θ) = p and (sec θ + cosec θ) = q, then q(p2 − 1) = ?
(a) 2
(b) 2p
(c)
(d)
Answer:
(b) 2p
Page No 326:
Question 23:
If (cos θ + sin θ) = cos θ, then (cos θ − sin θ) = ?
(a) sin θ
(b) sec θ
(c) cosec θ
(d) None of these
Answer:
(a) sin θ
Page No 326:
Question 24:
(a) tan θ
(b) cot θ
(c) sec θ
(d) cosec θ
Answer:
(a) tan θ
Page No 326:
Question 25:
(cosec θ − cot θ)2 = ?
(a)
(b)
(c)
(d)
Answer:
(b)
Page No 326:
Question 26:
If tan θ = then
(a)
(b)
(c)
(d)
Answer:
(c)
Page No 326:
Question 27:
If (sin θ + cos θ) = , then (tan θ + cot θ) = ?
(a)
(b)
(c) 1
(d)
Answer:
(c) 1
Page No 326:
Question 28:
If (sin θ + cos θ) = a and (sin3θ + cos3θ) = b, then (3a − 2b) = ?
(a) a3
(b) b3
(c) 0
(d) 1
Answer:
(a) a3
View NCERT Solutions for all chapters of Class 10