Prove the following identities
Question 1:
Answer:
LHS
RHS. Hence proved.
Question 2:
Answer:
LHS
RHS. Hence proved.
Question 3:
Answer:
LHS
RHS. Hence proved.
Question 4:
Answer:
LHS
RHS. Hence proved.
Question 5:
Answer:
LHS
RHS. Hence proved.
Question 6:
Answer:
LHS
RHS. Hence proved.
Question 7:
Answer:
LHS
RHS. Hence proved.
Question 8:
Answer:
LHS
RHS. Hence proved.
Question 9:
Answer:
LHS
RHS. Hence proved.
Question 10:
Answer:
LHS
RHS. Hence proved.
Question 11:
Answer:
LHS
RHS. Hence proved.
Question 12:
Answer:
LHS
RHS. Hence proved.
Question 13:
Answer:
LHS
RHS. Hence proved.
Question 14:
Answer:
RHS
LHS. Hence proved.
Question 15:
Answer:
LHS
RHS. Hence proved.
Question 16:
Answer:
LHS
Since
RHS. Hence proved.
Question 17:
Answer:
LHS
RHS. Hence proved.
Question 18:
Answer:
LHS
RHS. Hence proved.
Question 19:
Answer:
LHS
RHS. Hence proved.
Question 20:
Answer:
LHS
= RHS. Hence proved.
Question 21:
Answer:
LHS
RHS. Hence proved.
Question 22:
Answer:
LHS
RHS. Hence proved.
Question 23:
Answer:
LHS
RHS. Hence proved.
Question 24:
Answer:
LHS
RHS. Hence proved.
Question 25:
Answer:
LHS
RHS. Hence proved.
Question 26: Prove that:
Answer:
We know
RHS. Hence proved.
Question 27: Prove that
Answer:
LHS
We know,
Let
Since is in I quadrant, it is positive
RHS. Hence proved.
Question 28: i) If
and
lies in III quadrant, find the value of
,
and
ii) If
and
lies in II quadrant, find the value of
and
.
Answer:
i) Given
We know
Given that lies in II Quadrant,
is negative.
Similarly,
Given that lies in II Quadrant,
is positive.
Now
Given that lies in III Quadrant,
is negative.
Also
ii) Given
and
lies in II Quadrant
Since lies in II Quadrant,
is positive
We know
We know
Since lies in I Quadrant,
is positive
Question 29: If
and
lies in II quadrant, find the value of
,
and
Answer:
Given
Since lies in II Quadrant ,
is positive
We know
Since lies in I Quadrant ,
is positive
We know
Since lies in I Quadrant ,
is positive
Question 30:
i) if and
lie in II quadrant such that
, find the value of
,
and
ii) If
and
is acute, find
iii) If
and
, find the value of
Answer:
i) Given
Since lie in II Quadrant,
is negative
Now we know,
Since lies in II Quadrant,
will lie in I Quadrant. Hence
is positive.
Now
Since lies in II Quadrant,
will lie in I Quadrant. Hence
is positive.
ii) Given
iii)
and
Since likes in I Quadrant,
lies in II Quadrant
is negative
Now
Question 31: If
, find the value of
Answer:
Question 32: If
and If
, show that
Answer:
Given
and If
Hence
Question 33: Prove that
Answer:
LHS
RHS. Hence proved.
Question 34: Prove that
Answer:
LHS =
RHS. Hence proved.
Question 35: Prove that
Answer:
LHS =
RHS. Hence proved.
Question 36: Prove that
Answer:
LHS =
RHS. Hence proved.
Question 37: If , then prove that
Answer:
Given
LHS
RHS. Hence proved.
Question 38: If and
, prove that:
i)
ii)
Answer:
i) Given
… … … … … i)
Given
… … … … … ii)
Dividing i) by ii) we get
We know
ii) Squaring and adding
Question 39: If
, prove that
Answer:
Given
,
RHS
We know
Since
LHS. Hence proved.
Question 40: If
prove that
Answer:
Given
Applying Componendo and Dividendo
Question 41: If , prove that
Answer:
Given , prove that
. Hence proved.
Question 42:
and
, prove that
Answer:
Given
and
Squaring and adding the two
Since
Question 43: If
and
, prove that
Answer:
Given
Also given
Therefore
Question 44: If has
and
as its roots, then prove that:
i)
ii)
iii)
Answer:
i) Given … … … … … i)
… … … … … ii)
if and
are the roots of i) , then
and
are the roots of ii)
ii) Similarly
iii)
Question 45: If , then prove that
Answer:
Given
Squaring both sides