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