Question 1: If and , where , find the values of the following: i) ii) iii) iv)

Answer:

Given and , where

This means that A is in Quadrant I and B is in Quadrant I

Therefore,

Similarly,

i)

ii)

iii)

iv)

Question 2:

a) If and , where , find the values of the following: i) ii)

b) If and , where and both lie in Q II find the value of

Answer:

a) Given and , where

This means that A is in Quadrant II and B is in Quadrant I

Therefore,

Similarly,

i)

ii)

b) Given and , where and both lie in Q II

Therefore,

Similarly,

Question 3: If and , where , , find the values of the following: i) ii)

Answer:

Given and , where ,

This means that A is in Quadrant III and B is in Quadrant IV

Therefore,

Similarly,

i)

ii)

Question 4: If and , where and , find

Answer:

Given and , where and

This means that A is in Quadrant III and B is in Quadrant I

Therefore,

Question 5: If and , where and , find

Answer:

Given and , where and

This means that A is in Quadrant II and B is in Quadrant IV

Therefore,

Similarly,

Similarly,

Question 6: If and , where and , find the following: i) ii)

Answer:

Given and , where and

This means that A is in Quadrant II and B is in Quadrant I

Therefore,

Similarly,

Similarly,

i)

ii)

Question 7: Evaluate the following:

i) ii)

iii) iv)

Answer:

i)

ii)

iii)

iv)

Question 8: If and , where A lies in the second quadrant and B in the third quadrant, find the values of the following:

i) ii) iii)

Answer:

Given and , where A lies in the second quadrant and B in the third quadrant

Therefore,

i)

ii)

Question 9: Prove that

Answer:

LHS

RHS. Hence proved.

Question 10: Prove that

Answer:

LHS

RHS. Hence proved.

Question 11:

i)

ii)

iii)

Answer:

i)

ii)

iii)

Question 12:

i)

ii)

iii)

Answer:

i) LHS

Note:

RHS. Hence proved.

ii) LHS

Note:

RHS. Hence proved.

iii) LHS

Note:

RHS. Hence proved.

Question 13: Prove that:

Answer:

LHS

Note: Since

RHS. Hence proved.

Question 14: i) If and , prove that

ii) If and , prove that

Answer:

i)

ii)

Question 15: Prove that

i)

ii)

Answer:

i) LHS =

Lets calculate

Substituting it back

RHS. Hence proved.

ii) LHS

Since:

RHS. Hence Proved.

Question 16: 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) RHS

LHS. Hence Proved

v) LHS

RHS. Hence Proved.

vi) LHS

RHS. Hence proved.

Question 17: Prove that:

i)

ii)

iii)

iv)

Answer:

i)

ii)

iii)

iv)

Question 18: Prove that

Answer:

RHS

LHS. Hence proved.

Question 19: . show that

Answer:

Using Componendo and Dividendo

Question 20: If , prove that

Answer:

Now

Hence proved.

Question 21: If and , find the value of and

Answer:

Given and

Similarly,

Question 22: If and , prove that

Answer:

Given and

Hence proved.

Question 23: If and , prove that:

Answer:

Given and

Question 24: If lies in the first quadrant and , then prove that

Answer:

Given

Question 25: If , then prove that

Answer:

Given

. Hence proved.

Question 26: If and , where , then find the values of and

Answer:

Given:

… … … i)

… … … ii)

Solving i) and ii) we get

Also

Question 27: If and are two different values of lying between and which satisfy the equation , find the value of

Answer:

Given

Eliminating

Squaring both sides

If and are roots of equation

Similarly

Eliminating

Squaring both sides

If and are roots of equation

Now

Question 28: If and , show that

i)

ii)

Answer:

Given, and

Similarly,

Question 29: Prove that:

i)

ii)

iii)

Answer:

i) LHS

RHS. Hence proved.

ii) LHS

RHS. Hence proved.

iii) LHS

RHS. Hence proved.

Question 30: If , prove that

Answer:

Given

Hence Proved.

Question 31: If and , show that

Answer:

Given and

. Hence proved.

Question 32: If angle is divided into two parts such that the tangents of one parts is times the tangent of other, and is their difference, then show that

Answer:

Let and be the two parts

Applying componendo and dividendo

Hence proved.

Question 33: If , then show that

Answer:

Given

Dividing both numerator and denominator by

Hence proved.

Question 34: If and are two solutions of the equation , then find the values of and

Answer:

Given

Squaring both sides

If and are the roots then

Similarly

Squaring both sides

If and are the roots then

Now