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Trigonometric Identities

Reading Time: 15 mins

Solved Examples: 8

Practice Questions: 3

What are Trigonometric Identities?

Trigonometric Identities are equations involving trigonometric ratios that are true for all angles for which the expressions are defined. Unlike ordinary equations, which are true only for specific values of the variable, trigonometric identities remain valid regardless of the value of the angle, provided the ratios involved are defined. These identities establish relationships between the six trigonometric ratios—sin θ, cos θ, tan θ, cosec θ, sec θ, and cot θ

Reciprocal Identities

The reciprocal identities express one trigonometric ratio in terms of the reciprocal of another. These identities follow directly from the definitions of the six trigonometric ratios.

cosec θ = 1/sin θ
sin θ = 1/cosec θ
sec θ = 1/cos θ
cos θ = 1/sec θ
cot θ = 1/tan θ
tan θ = 1/cot θ

These identities are useful for converting one trigonometric ratio into another and for simplifying expressions.

Quotient Identities

The quotient identities relate tan θ and cot θ to sin θ and cos θ.
Formulae:

tan θ = sin θ/cos θ
cot θ = cos θ/sin θ

Pythagorean Identities

The Pythagorean identities are the most important trigonometric identities. They are derived from the Pythagorean theorem and establish relationships among sin θ, cos θ, tan θ, sec θ, cot θ and cosec θ.

sin² θ + cos² θ = 1
1 + tan² θ = sec² θ
1 + cot² θ = cosec² θ

Difference Between an Identity and an Equation

A trigonometric identity is true for every value of θ for which the expressions are defined.
Example:
sin² θ + cos² θ = 1
This identity is always true.
A trigonometric equation is true only for specific values of θ.
Example:
sin θ = ½
This equation is true only for particular angles such as 30° and 150° (within 0° to 360°).

Summary of Trigonometric Identities

Identity	Formula
Reciprocal Identity (sin θ)	cosec θ = 1/sin θ, sin θ = 1/cosec θ
Reciprocal Identity (cos θ)	sec θ = 1/cos θ, cos θ = 1/sec θ
Reciprocal Identity (tan θ)	cot θ = 1/tan θ, tan θ = 1/cot θ
Quotient Identity	tan θ = sin θ/cos θ, cot θ = cos θ/sin θ
Pythagorean Identity	sin² θ + cos² θ = 1
Pythagorean Identity	1 + tan² θ = sec² θ
Pythagorean Identity	1 + cot² θ = cosec² θ
Rearranged Form	sin² θ = 1 − cos² θ, cos² θ = 1 − sin² θ
Rearranged Form	tan² θ = sec² θ − 1, sec² θ = 1 + tan² θ
Rearranged Form	cot² θ = cosec² θ − 1, cosec² θ = 1 + cot² θ

Solved Examples

Question 1: If tan θ = 3/4, find the value of (tan θ + cot θ)/(tan θ × cot θ)

Using the reciprocal identity, cot θ = 4/3
Numerator
= 3/4 + 4/3
= (9 + 16)/12
= 25/12
Denominator
= 3/4 × 4/3
= 1
Therefore,
= 25/12

Question 2: If cosec θ = 25/24, find the value of (1 + sin θ)/(1 − sin θ)

Using the reciprocal identity,
sin θ = 24/25
Therefore,
= (1 + 24/25)/(1 − 24/25)
= (49/25)/(1/25)
= 49

Question 3: If sec θ = 5/3 and tan θ = 4/3, find the value of (1/sec θ) + (1/tan θ)

Using reciprocal identities,
1/sec θ = cos θ = 3/5
1/tan θ = cot θ = 3/4
Therefore,
= 3/5 + 3/4
= (12 + 15)/20
= 27/20

Question 4: If sin θ = 5/13 and cos θ = 12/13, find the value of: tan θ + cot θ

tan θ = (5/13)/(12/13)
= 5/12
cot θ = (12/13)/(5/13)
= 12/5
Therefore,
tan θ + cot θ
= 5/12 + 12/5
= (25 + 144)/60
= 169/60

Question 5: If sin θ = 3/5 and θ is an acute angle, find cos θ.

Using the identity,
sin² θ + cos² θ = 1
cos² θ = 1 − (3/5)²
= 1 − 9/25
= 16/25
Since θ is acute,
cos θ = 4/5

Question 6: If tan θ = 3/4, find sec θ.

Using the identity,
1 + tan² θ = sec² θ
sec² θ
= 1 + (3/4)²
= 1 + 9/16
= 25/16
Therefore,
sec θ = 5/4

Question 7: Simplify (sec² θ − tan² θ) + (cosec² θ − cot² θ)

Using the identities,
sec² θ − tan² θ = 1
cosec² θ − cot² θ = 1
Therefore,
= 1 + 1
= 2

Question 8: Simplify (1 + cot² θ)(1 − sin² θ)

Using the identities,
1 + cot² θ = cosec² θ
1 − sin² θ = cos² θ
Therefore,
= cosec² θ × cos² θ
= (1/sin² θ) × cos² θ
= cot² θ

Question 9: Simplify (cosec² θ − cot² θ)(sec² θ − tan² θ)

Using the identities,
cosec² θ − cot² θ = 1
sec² θ − tan² θ = 1
Therefore,
= 1 × 1
= 1

Question 10: Simplify (sec² θ + cosec² θ) − (tan² θ + cot² θ)

Using the identities,
sec² θ = 1 + tan² θ
cosec² θ = 1 + cot² θ
Therefore,
= (1 + tan² θ) + (1 + cot² θ) − tan² θ − cot² θ
= 2

Question 11: Verify that (sec² θ − 1)(cosec² θ − 1) = 1

LHS
= (sec² θ − 1)(cosec² θ − 1)
Using,
sec² θ − 1 = tan² θ
cosec² θ − 1 = cot² θ
Therefore,
= tan² θ × cot² θ
= (tan θ × cot θ)²
= 1²
= 1
= RHS

Question 12: Verify that (1 − cos² θ)(1 + tan² θ) = tan² θ

LHS
= (1 − cos² θ)(1 + tan² θ)
Using,
1 − cos² θ = sin² θ
1 + tan² θ = sec² θ
Therefore,
= sin² θ × sec² θ
= sin² θ/cos² θ
= tan² θ
= RHS
Hence verified.

Question 13: If cos θ = 12/13 and θ is an acute angle, find the value of cosec θ − cot θ

Using,
sin² θ = 1 − cos² θ
= 1 − (12/13)²
= 25/169
∴ sin θ = 5/13
Hence,
cosec θ = 13/5
cot θ = 12/5
Therefore,
cosec θ − cot θ
= 13/5 − 12/5
= 1/5

Question 14: If cosec θ = 25/24 and θ is an acute angle, find the value of sec θ − tan θ

sin θ = 24/25
Using, cos² θ = 1 − (24/25)²
= 49/625
∴ cos θ = 7/25
Hence,
sec θ = 25/7
tan θ = (24/25)/(7/25)
= 24/7
Therefore,
sec θ − tan θ
= 25/7 − 24/7
= 1/7

Question 15: If cos θ = 4/5 and θ is an acute angle, find the value of (1 − sin² θ)sec² θ

Using,
1 − sin² θ = cos² θ
Therefore,
= cos² θ × sec² θ
= 1

Common Mistakes

Confusing Identities with Equations:
Students often treat a trigonometric identity as if it is true only for specific values of θ. Remember that an identity is true for every value of θ for which the expressions are defined, whereas an equation is true only for particular values of θ.
Using the Wrong Identity:
Students sometimes apply an incorrect identity while simplifying an expression. Always identify whether the problem requires a reciprocal, quotient or Pythagorean identity before substituting.
Formulae:
sin² θ + cos² θ = 1
1 + tan² θ = sec² θ
1 + cot² θ = cosec² θ
Forgetting the Rearranged Forms:
Students often remember the basic Pythagorean identities but forget their rearranged forms. These are frequently needed to find an unknown trigonometric ratio.
Formulae:
sin² θ = 1 − cos² θ
cos² θ = 1 − sin² θ
tan² θ = sec² θ − 1
cot² θ = cosec² θ − 1
Confusing Reciprocal and Quotient Identities:
Students sometimes assume that tan θ = cos θ/sin θ or cot θ = sin θ/cos θ. Remember that tan θ = sin θ/cos θ and cot θ = cos θ/sin θ, while sec θ, cosec θ and cot θ are reciprocal ratios.
Not Simplifying Step by Step:
Students often substitute several identities at once, making algebraic mistakes. Replace only one identity at a time, simplify the expression after each substitution, and then apply the next identity if required. This reduces errors and makes the solution easier to verify.

Practice Questions

Question 1: If sin θ = 5/13 and θ is an acute angle, find the value of (1 + tan² θ)(1 − sin² θ)

Question 2: Simplify (sec² θ − tan² θ) + (sin² θ + cos² θ) + (cosec² θ − cot² θ)

Question 3: If tan θ = 12/5 and θ is an acute angle, find the value of (sec² θ − 1)/(1 − cos² θ)

Question 4: Simplify [(1 + tan² θ)(1 − sin² θ)] + [(1 + cot² θ)(1 − cos² θ)]

Question 5: If sin θ = 15/17 and θ is an acute angle, find the value of [(1 − cos² θ) + (sec² θ − 1)] ÷ tan² θ