Trigonometrical Identity

Definition of trigonometrical identity:

An equation which is true for all values of the variable involved is called an identity. An equation which involves trigonometric ratios of an angle and is true for all the values of the angle is called trigonometrical identities.

When the solutions of any trigonometric ratio problems represent the same expression in the L.H.S. and R.H.S. and the relation is satisfied for all the values of θ then such relation is called a trigonometrical identity.

Mutual relations among the trigonometrical ratios are generally used to establish the equality of such trigonometrical identities.

To solve different types of trignometrical identity follow the formula:

● sin θ ∙ csc θ = 1 ⇒  csc θ = 1/sin θ

● cos θ ∙ sec θ = 1 ⇒ sec θ = 1/cos θ 

● tan θ ∙ cot θ = 1 ⇒ cot θ = 1/tan θ 

● tan θ = sin θ/cos θ                 

● cot θ = cos θ/sin θ

● sin² θ implies (sin θ)²
similarly, tan³ θ means (tan θ)³ etc.

● sin² θ + cos² θ = 1

cos² θ = 1 - sin² θ
sin² θ = 1 - cos² θ

● sec² θ = 1 + tan² θ
sec² θ - tan² θ = 1
tan² θ = sec² θ - 1

● csc² θ = 1 + cot² θ
csc² θ - 1 = cot² θ
csc² θ - cot² θ = 1

● The trigonometrical ratios of a positive acute angle θ are always non-negative and

(i) sin θ and cos θ can never be greater than 1;

(ii) sec θ and csc θ can never be less than 1;

(iii) tan θ and cot θ can have any value.

Worked-out problems on trigonometric identity:

1. Proof the identity:

tan² θ – (1/cos² θ) + 1 = 0

Solution:

L.H.S = tan² θ – (1/cos² θ) + 1

= tan² θ - sec² θ + 1 [since, 1/cos θ = sec θ]

= tan² θ – (1 + tan² θ) +1 [since, sec² θ = 1 + tan² θ]

= tan² θ – 1 – tan² θ + 1

= 0 = R.H.S. Proved


2. Verify that:

1/(sin θ + cos θ) + 1/(sin θ - cos θ) = 2 sin θ/(1 – 2 cos² θ)

Solution:

L.H.S = 1/(sin θ + cos θ) + 1/(sin θ - cos θ)

= [(sin θ - cos θ) + (sin θ + cos θ)]/(sin θ + cos θ)(sin θ - cos θ)

= [sin θ - cos θ + sin θ + cos θ]/(sin² θ - cos² θ)

= 2 sin θ/[(1 - cos² θ) - cos² θ] [since, sin² θ = 1 - cos² θ]

= 2 sin θ/[1 - cos² θ - cos² θ]

= 2 sin θ/[1 – 2 cos² θ] = R.H.S. Proved


3. Prove that:

sec² θ + csc² θ = sec² θ ∙ csc² θ

Solution:

L.H.S. = sec² θ + csc² θ

= 1/cos² θ + 1/sin² θ [since, sec θ = 1/cos θ and csc θ = 1/sin θ]

= (sin² θ + cos² θ)/(cos² θ sin² θ)

= 1/cos² θ ∙ sin² θ [since, sin² θ + cos² θ = 1]

= 1/cos² θ ∙ 1/sin² θ

= sec² θ ∙ csc² θ = R.H.S. Proved




More examples on trigonometrical identity are explained below. To proof the identities step-by-step follow the above trig formulas.

4. Prove the identity:

cos θ/(1 + sin θ) = (1 + cos θ - sin θ)/(1 + cos θ + sin θ)

Solution:

R. H. S. = (1 + cos θ - sin θ)/(1 + cos θ + sin θ)

= {(1 + cos θ - sin θ) (1 + cos θ + sin θ)}/{(1+ cos θ + sin θ) (1 + cos θ + sin θ)} [multiplying both numerator and denominator by (1 + cos θ + sin θ)]

= {(1 + cos θ)² - sin² θ}/(1 + cos θ + sin θ)²

= (1 + cos² θ + 2 cos θ - sin² θ)/{(1 + cos θ)² + 2 ∙ (1 + cos θ) sin θ + sin² θ}

= (cos² θ + 2 cos θ + 1 - sin² θ)/{1 + cos² θ + 2 cos θ + 2 ∙ (1 + cos θ) ∙ sin θ + sin² θ}

= (cos² θ + 2 cos θ + cos² θ)/{2 + 2 cos θ + 2 ∙ (1 + cos θ) ∙ sin θ} [since, sin² θ + cos² θ = 1 and 1 - sin² θ = cos² θ]

= {2 cos θ (1 + cos θ)}/{2 (1 + cos θ)(1 + sin θ)}

= cos θ/(1 + sin θ) = L.H.S. Proved


5. Verify the trigonometrical identity:

(cot θ + csc θ – 1)/(cot θ - csc θ + 1) = (1 + cos θ)/sin θ

L.H.S. = (cot θ + csc θ – 1)/(cot θ - csc θ + 1)

= {cot θ + csc θ - (csc² θ - cot² θ)}/(cot θ - csc θ + 1)

[csc² θ = 1 + cot² θ ⇒ csc² θ - cot² θ = 1]

= {(cot θ + csc θ) - (csc θ + cot θ) (csc θ - cot θ)}/(cot θ - csc θ + 1)

= {(cot θ + csc θ) (1 - csc θ + cot θ)}/ (1 - csc θ + cot θ)

= cot θ + csc θ

= (cos θ/sin θ) + (1/sin θ)

= (1 + cos θ)/sin θ = R.H.S. Proved

● Trigonometric Functions

• Basic Trigonometric Ratios and Their Names
• Restrictions of Trigonometrical Ratios
• Reciprocal Relations of Trigonometric Ratios
• Quotient Relations of Trigonometric Ratios
  
• Limit of Trigonometric Ratios
  
• Trigonometrical Identity
• Problems on Trigonometric Identities
• Elimination of Trigonometric Ratios
• Eliminate Theta between the equations
• Problems on Eliminate Theta
• Trig Ratio Problems
• Proving Trigonometric Ratios
• Trig Ratios Proving Problems
• Verify Trigonometric Identities
• Trigonometrical Ratios of 0°
• Trigonometrical Ratios of 30°
• Trigonometrical Ratios of 45°
• Trigonometrical Ratios of 60°
• Trigonometrical Ratios of 90°
• Trigonometrical Ratios Table
• Problems on Trigonometric Ratio of Standard Angle
  
• Trigonometrical Ratios of Complementary Angles
• Rules of Trigonometric Signs
• Signs of Trigonometrical Ratios
  
• All Sin Tan Cos Rule
• Trigonometrical Ratios of (- θ)
• Trigonometrical Ratios of (90° + θ)
• Trigonometrical Ratios of (90° - θ)
• Trigonometrical Ratios of (180° + θ)
• Trigonometrical Ratios of (180° - θ)
• Trigonometrical Ratios of (270° + θ)
• Trigonometrical Ratios of (270° - θ)
• Trigonometrical Ratios of (360° + θ)
• Trigonometrical Ratios of (360° - θ)
• Trigonometrical Ratios of any Angle
• Trigonometrical Ratios of some Particular Angles
• Trigonometric Ratios of an Angle
• Trigonometric Functions of any Angles
• Problems on Trigonometric Ratios of an Angle
• Problems on Signs of Trigonometrical Ratios

10th Grade Math

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