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 Page 1


5. Trigonometric Functions
Exercise 5.1
1. Question
Prove the following identities
sec
4
x – sec
2
x = tan
4
x + tan
2
x
Answer
LHS = sec
4
x – sec
2
x
= (sec
2
x) 
2
 – sec
2
x
We know sec
2
 ? = 1 + tan
2
 ?.
= (1 + tan
2
x) 
2
 – (1 + tan
2
x)
= 1 + 2tan
2
x + tan
4
x – 1 - tan
2
x
= tan
4
x + tan
2
x = RHS
Hence proved.
2. Question
Prove the following identities
sin
6
x + cos
6
x = 1 – 3 sin
2
x cos
2
x
Answer
LHS = sin
6
x + cos
6
x
= (sin
2
x)
3
 + (cos
2
x)
3
We know that a
3
 + b
3
 = (a + b) (a
2
 + b
2
 – ab)
= (sin
2
x + cos
2
x) [(sin
2
x)
2
 + (cos
2
x)
2
 – sin
2
x cos
2
x]
We know that sin
2
x + cos
2
x = 1 and a
2
 + b
2
 = (a + b)
2
 – 2ab
= 1 × [(sin
2
x + cos
2
x)
2
 – 2sin
2
x cos
2
x – sin
2
x cos
2
x
= 1
2
 - 3sin
2
x cos
2
x
= 1 - 3sin
2
x cos
2
x = RHS
Hence proved.
3. Question
Prove the following identities
(cosecx – sinx) (secx – cosx) (tanx + cotx) = 1
Answer
LHS = (cosecx – sinx) (secx – cosx) (tanx + cotx)
We know that 
Page 2


5. Trigonometric Functions
Exercise 5.1
1. Question
Prove the following identities
sec
4
x – sec
2
x = tan
4
x + tan
2
x
Answer
LHS = sec
4
x – sec
2
x
= (sec
2
x) 
2
 – sec
2
x
We know sec
2
 ? = 1 + tan
2
 ?.
= (1 + tan
2
x) 
2
 – (1 + tan
2
x)
= 1 + 2tan
2
x + tan
4
x – 1 - tan
2
x
= tan
4
x + tan
2
x = RHS
Hence proved.
2. Question
Prove the following identities
sin
6
x + cos
6
x = 1 – 3 sin
2
x cos
2
x
Answer
LHS = sin
6
x + cos
6
x
= (sin
2
x)
3
 + (cos
2
x)
3
We know that a
3
 + b
3
 = (a + b) (a
2
 + b
2
 – ab)
= (sin
2
x + cos
2
x) [(sin
2
x)
2
 + (cos
2
x)
2
 – sin
2
x cos
2
x]
We know that sin
2
x + cos
2
x = 1 and a
2
 + b
2
 = (a + b)
2
 – 2ab
= 1 × [(sin
2
x + cos
2
x)
2
 – 2sin
2
x cos
2
x – sin
2
x cos
2
x
= 1
2
 - 3sin
2
x cos
2
x
= 1 - 3sin
2
x cos
2
x = RHS
Hence proved.
3. Question
Prove the following identities
(cosecx – sinx) (secx – cosx) (tanx + cotx) = 1
Answer
LHS = (cosecx – sinx) (secx – cosx) (tanx + cotx)
We know that 
We know that sin
2
x + cos
2
x = 1.
= 1 = RHS
Hence proved.
4. Question
Prove the following identities
cosecx (secx – 1) – cotx (1 – cosx) = tanx – sinx
Answer
LHS = cosecx (secx – 1) – cotx (1 – cosx)
We know that 
We know that 1 – cos
2
x = sin
2
x.
= RHS
Hence proved.
5. Question
Prove the following identities
Answer
LHS 
We know that 
Page 3


5. Trigonometric Functions
Exercise 5.1
1. Question
Prove the following identities
sec
4
x – sec
2
x = tan
4
x + tan
2
x
Answer
LHS = sec
4
x – sec
2
x
= (sec
2
x) 
2
 – sec
2
x
We know sec
2
 ? = 1 + tan
2
 ?.
= (1 + tan
2
x) 
2
 – (1 + tan
2
x)
= 1 + 2tan
2
x + tan
4
x – 1 - tan
2
x
= tan
4
x + tan
2
x = RHS
Hence proved.
2. Question
Prove the following identities
sin
6
x + cos
6
x = 1 – 3 sin
2
x cos
2
x
Answer
LHS = sin
6
x + cos
6
x
= (sin
2
x)
3
 + (cos
2
x)
3
We know that a
3
 + b
3
 = (a + b) (a
2
 + b
2
 – ab)
= (sin
2
x + cos
2
x) [(sin
2
x)
2
 + (cos
2
x)
2
 – sin
2
x cos
2
x]
We know that sin
2
x + cos
2
x = 1 and a
2
 + b
2
 = (a + b)
2
 – 2ab
= 1 × [(sin
2
x + cos
2
x)
2
 – 2sin
2
x cos
2
x – sin
2
x cos
2
x
= 1
2
 - 3sin
2
x cos
2
x
= 1 - 3sin
2
x cos
2
x = RHS
Hence proved.
3. Question
Prove the following identities
(cosecx – sinx) (secx – cosx) (tanx + cotx) = 1
Answer
LHS = (cosecx – sinx) (secx – cosx) (tanx + cotx)
We know that 
We know that sin
2
x + cos
2
x = 1.
= 1 = RHS
Hence proved.
4. Question
Prove the following identities
cosecx (secx – 1) – cotx (1 – cosx) = tanx – sinx
Answer
LHS = cosecx (secx – 1) – cotx (1 – cosx)
We know that 
We know that 1 – cos
2
x = sin
2
x.
= RHS
Hence proved.
5. Question
Prove the following identities
Answer
LHS 
We know that 
We know that a
3
 + b
3
 = (a + b) (a
2
 + b
2
 – ab)
We know that sin
2
x + cos
2
x = 1.
= sinx
= RHS
Hence proved.
6. Question
Prove the following identities
 = (secx cosecx + 1)
Answer
LHS 
We know that 
We know that a
3
 - b
3
 = (a - b) (a
2
 + b
2
 + ab)
We know that sin
2
x + cos
2
x = 1.
Page 4


5. Trigonometric Functions
Exercise 5.1
1. Question
Prove the following identities
sec
4
x – sec
2
x = tan
4
x + tan
2
x
Answer
LHS = sec
4
x – sec
2
x
= (sec
2
x) 
2
 – sec
2
x
We know sec
2
 ? = 1 + tan
2
 ?.
= (1 + tan
2
x) 
2
 – (1 + tan
2
x)
= 1 + 2tan
2
x + tan
4
x – 1 - tan
2
x
= tan
4
x + tan
2
x = RHS
Hence proved.
2. Question
Prove the following identities
sin
6
x + cos
6
x = 1 – 3 sin
2
x cos
2
x
Answer
LHS = sin
6
x + cos
6
x
= (sin
2
x)
3
 + (cos
2
x)
3
We know that a
3
 + b
3
 = (a + b) (a
2
 + b
2
 – ab)
= (sin
2
x + cos
2
x) [(sin
2
x)
2
 + (cos
2
x)
2
 – sin
2
x cos
2
x]
We know that sin
2
x + cos
2
x = 1 and a
2
 + b
2
 = (a + b)
2
 – 2ab
= 1 × [(sin
2
x + cos
2
x)
2
 – 2sin
2
x cos
2
x – sin
2
x cos
2
x
= 1
2
 - 3sin
2
x cos
2
x
= 1 - 3sin
2
x cos
2
x = RHS
Hence proved.
3. Question
Prove the following identities
(cosecx – sinx) (secx – cosx) (tanx + cotx) = 1
Answer
LHS = (cosecx – sinx) (secx – cosx) (tanx + cotx)
We know that 
We know that sin
2
x + cos
2
x = 1.
= 1 = RHS
Hence proved.
4. Question
Prove the following identities
cosecx (secx – 1) – cotx (1 – cosx) = tanx – sinx
Answer
LHS = cosecx (secx – 1) – cotx (1 – cosx)
We know that 
We know that 1 – cos
2
x = sin
2
x.
= RHS
Hence proved.
5. Question
Prove the following identities
Answer
LHS 
We know that 
We know that a
3
 + b
3
 = (a + b) (a
2
 + b
2
 – ab)
We know that sin
2
x + cos
2
x = 1.
= sinx
= RHS
Hence proved.
6. Question
Prove the following identities
 = (secx cosecx + 1)
Answer
LHS 
We know that 
We know that a
3
 - b
3
 = (a - b) (a
2
 + b
2
 + ab)
We know that sin
2
x + cos
2
x = 1.
We know that 
= cosecx × secx + 1
= secx cosecx + 1
= RHS
Hence proved.
7. Question
Prove the following identities
Answer
LHS 
We know that a
3
 ± b
3
 = (a ± b) (a
2
 + b
2
± ab)
We know that sin
2
x + cos
2
x = 1.
= 1 - sinx cosx + 1 + sinx cosx
= 2
= RHS
Hence proved.
8. Question
Prove the following identities
(secx sec y + tanx tan y)
2
 – (secx tan y + tanx sec y)
2
 = 1
Answer
LHS = (secx sec y + tanx tan y)
2
 – (secx tan y + tanx sec y)
2
= [(secx sec y)
2
 + (tanx tan y)
2
 + 2 (secx sec y) (tanx tan y)] – [(secx tan y)
2
 + (tanx sec y)
2
 + 2 (secx tan y)
(tanx sec y)]
= [sec
2
x sec
2
 y + tan
2
x tan
2
 y + 2 (secx sec y) (tanx tan y)] – [sec
2
x tan
2
 y + tan
2
x sec
2
 y + 2 (sec
2
x tan
2
 y)
(tanx sec y)]
= sec
2
x sec
2
 y - sec
2
x tan
2
 y + tan
2
x tan
2
 y - tan
2
x sec
2
 y
= sec
2
x (sec
2
 y - tan
2
 y) + tan
2
x (tan
2
 y - sec
2
 y)
= sec
2
x (sec
2
 y - tan
2
 y) - tan
2
x (sec
2
 y - tan
2
 y)
We know that sec
2
x – tan
2
x = 1.
= sec
2
x × 1 – tan
2
x × 1
= sec
2
x – tan
2
x
= 1
Page 5


5. Trigonometric Functions
Exercise 5.1
1. Question
Prove the following identities
sec
4
x – sec
2
x = tan
4
x + tan
2
x
Answer
LHS = sec
4
x – sec
2
x
= (sec
2
x) 
2
 – sec
2
x
We know sec
2
 ? = 1 + tan
2
 ?.
= (1 + tan
2
x) 
2
 – (1 + tan
2
x)
= 1 + 2tan
2
x + tan
4
x – 1 - tan
2
x
= tan
4
x + tan
2
x = RHS
Hence proved.
2. Question
Prove the following identities
sin
6
x + cos
6
x = 1 – 3 sin
2
x cos
2
x
Answer
LHS = sin
6
x + cos
6
x
= (sin
2
x)
3
 + (cos
2
x)
3
We know that a
3
 + b
3
 = (a + b) (a
2
 + b
2
 – ab)
= (sin
2
x + cos
2
x) [(sin
2
x)
2
 + (cos
2
x)
2
 – sin
2
x cos
2
x]
We know that sin
2
x + cos
2
x = 1 and a
2
 + b
2
 = (a + b)
2
 – 2ab
= 1 × [(sin
2
x + cos
2
x)
2
 – 2sin
2
x cos
2
x – sin
2
x cos
2
x
= 1
2
 - 3sin
2
x cos
2
x
= 1 - 3sin
2
x cos
2
x = RHS
Hence proved.
3. Question
Prove the following identities
(cosecx – sinx) (secx – cosx) (tanx + cotx) = 1
Answer
LHS = (cosecx – sinx) (secx – cosx) (tanx + cotx)
We know that 
We know that sin
2
x + cos
2
x = 1.
= 1 = RHS
Hence proved.
4. Question
Prove the following identities
cosecx (secx – 1) – cotx (1 – cosx) = tanx – sinx
Answer
LHS = cosecx (secx – 1) – cotx (1 – cosx)
We know that 
We know that 1 – cos
2
x = sin
2
x.
= RHS
Hence proved.
5. Question
Prove the following identities
Answer
LHS 
We know that 
We know that a
3
 + b
3
 = (a + b) (a
2
 + b
2
 – ab)
We know that sin
2
x + cos
2
x = 1.
= sinx
= RHS
Hence proved.
6. Question
Prove the following identities
 = (secx cosecx + 1)
Answer
LHS 
We know that 
We know that a
3
 - b
3
 = (a - b) (a
2
 + b
2
 + ab)
We know that sin
2
x + cos
2
x = 1.
We know that 
= cosecx × secx + 1
= secx cosecx + 1
= RHS
Hence proved.
7. Question
Prove the following identities
Answer
LHS 
We know that a
3
 ± b
3
 = (a ± b) (a
2
 + b
2
± ab)
We know that sin
2
x + cos
2
x = 1.
= 1 - sinx cosx + 1 + sinx cosx
= 2
= RHS
Hence proved.
8. Question
Prove the following identities
(secx sec y + tanx tan y)
2
 – (secx tan y + tanx sec y)
2
 = 1
Answer
LHS = (secx sec y + tanx tan y)
2
 – (secx tan y + tanx sec y)
2
= [(secx sec y)
2
 + (tanx tan y)
2
 + 2 (secx sec y) (tanx tan y)] – [(secx tan y)
2
 + (tanx sec y)
2
 + 2 (secx tan y)
(tanx sec y)]
= [sec
2
x sec
2
 y + tan
2
x tan
2
 y + 2 (secx sec y) (tanx tan y)] – [sec
2
x tan
2
 y + tan
2
x sec
2
 y + 2 (sec
2
x tan
2
 y)
(tanx sec y)]
= sec
2
x sec
2
 y - sec
2
x tan
2
 y + tan
2
x tan
2
 y - tan
2
x sec
2
 y
= sec
2
x (sec
2
 y - tan
2
 y) + tan
2
x (tan
2
 y - sec
2
 y)
= sec
2
x (sec
2
 y - tan
2
 y) - tan
2
x (sec
2
 y - tan
2
 y)
We know that sec
2
x – tan
2
x = 1.
= sec
2
x × 1 – tan
2
x × 1
= sec
2
x – tan
2
x
= 1
= RHS
Hence proved.
9. Question
Prove the following identities
Answer
RHS 
We know that sin
2
x + cos
2
x = 1.
We know that 1 – cos
2
x = sin
2
x.
We know that 1 – sin
2
x = cos
2
x.

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