Question 1: Express each of the following as the product of sines and cosines:
i) ii)
iii)
iv) v)
Answer:
i)
ii)
iii)
iv)
v)
Question 2: Prove that:
i) ii)
iii) iv)
v) vi)
Answer:
i) LHS
RHS. Hence proved.
ii) LHS
RHS. Hence proved.
iii) LHS
RHS. Hence proved.
iv) LHS
RHS. Hence proved.
v) LHS
RHS. Hence proved.
vi) LHS
RHS. Hence proved.
Question 3: Prove that:
i) ii)
iii) v)
v)
vi)
vii) viii)
Answer:
i) LHS
RHS. Hence proved.
ii) LHS
RHS. Hence proved.
iii) LHS
RHS. Hence proved.
iv) LHS
RHS. Hence proved.
v) LHS
RHS. Hence proved.
vi) LHS
RHS. Hence proved.
vii) LHS
RHS. Hence proved.
viii) LHS
RHS. Hence proved.
Question 4: Prove that:
i)
ii)
Answer:
i) LHS
RHS. Hence proved.
ii) LHS
RHS. Hence proved.
Question 5: Prove that:
i) ii)
Answer:
i) LHS
RHS. Hence proved.
ii) LHS
LHS
RHS. Hence proved.
Question 6: Prove that:
i)
ii)
iii)
iv)
v)
vi)
vii)
Answer:
i) LHS
RHS. Hence proved.
ii) LHS
RHS. Hence proved.
iii) LHS
RHS. Hence proved.
iv) LHS
RHS. Hence proved.
v) LHS
RHS. Hence proved.
vi) LHS
RHS. Hence proved.
vii) LHS
RHS. Hence proved.
Question 7: Prove that:
i)
ii)
iii)
iv)
v)
Answer:
i) LHS
RHS. Hence proved.
ii) LHS
RHS. Hence proved.
iii) LHS
RHS. Hence proved.
iv) LHS
RHS. Hence proved.
v) LHS
RHS. Hence proved.
Question 8: Prove that:
i)
ii)
iii)
iv)
v)
vi)
vii)
viii)
ix)
x)
xi)
Answer:
i) LHS
RHS. Hence proved.
ii) LHS
RHS. Hence proved.
iii) LHS
RHS. Hence proved.
iv) LHS
RHS. Hence proved.
v) LHS
RHS. Hence proved.
vi) LHS
RHS
vii) LHS
RHS
viii) LHS
RHS
ix) LHS
RHS
x) LHS
RHS. Hence proved.
xi) LHS
RHS. Hence proved.
Question 9: Prove that:
i)
ii)
Answer:
i) LHS
RHS. Hence proved.
ii) LHS
RHS. Hence proved.
Question 10: If
and
, prove that:
Answer:
Given:
and
Dividing one by another
Question 11: If , then prove that
Answer:
Given:
Question 12: If , prove that
Answer:
Given
Applying componendo and dividendo
Hence proved.
Question 13: Prove that:
i)
ii)
Answer:
i) LHS
ii) LHS
RHS. Hence proved.
Question 14: If
, prove that
Answer:
Given
Dividing both sides by
Hence proved.
Question 15: If , then prove that
Answer:
Given
Dividing both sides by
Hence proved.
Question 16: If , prove that
Answer:
Given
Applying componendo and dividendo
Hence proved.
Question 17: If , prove that
Answer:
Given
Divide both sides by we get
Hence proved.
Question 18: If
, prove that
Answer:
Given
RHS. Hence proved.
Question 19: If , prove that
Answer:
Given:
Applying componendo and dividendo
Hence proved.