Problems on Trigonometric Identities

Here we will prove the problems on trigonometric identities. In an identity there are two sides of the equation, one side is known as ‘left hand side’ and the other side is known as ‘right hand side’ and to prove the identity we need to use logical steps showing that one side of the equation ends up with the other side of the equation.

Proving the problems on trigonometric identities:

1. (1 - sin A)/(1 + sin A) = (sec A - tan A)²

Solution:

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

= (1 - sin A)²/(1 - sin A) (1 + sin A),[Multiply both numerator and denominator by (1 - sin A)



= (1 - sin A)²/(1 - sin² A)

= (1 - sin A)²/(cos² A), [Since sin² θ + cos² θ = 1 ⇒ cos² θ = 1 - sin² θ]

= {(1 - sin A)/cos A}²

= (1/cos A - sin A/cos A)²

= (sec A – tan A)² = R.H.S. Proved.


2. Prove that, √{(sec θ – 1)/(sec θ + 1)} = cosec θ - cot θ.

Solution:

L.H.S.= √{(sec θ – 1)/(sec θ + 1)}

= √[{(sec θ - 1) (sec θ - 1)}/{(sec θ + 1) (sec θ - 1)}]; [multiplying numerator and denominator by (sec θ - l) under radical sign]

= √{(sec θ - 1)²/(sec² θ - 1)}

=√{(sec θ -1)²/tan² θ}; [since, sec² θ = 1 + tan² θ ⇒ sec² θ - 1 = tan² θ]

= (sec θ – 1)/tan θ

= (sec θ/tan θ) – (1/tan θ)

= {(1/cos θ)/(sin θ/cos θ)} - cot θ

= {(1/cos θ) × (cos θ/sin θ)} - cot θ

= (1/sin θ) - cot θ

= cosec θ - cot θ = R.H.S. Proved.


3. tan⁴ θ + tan² θ = sec⁴ θ - sec² θ

Solution:

L.H.S = tan⁴ θ + tan² θ

= tan² θ (tan² θ + 1)

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

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

= sec⁴ θ - sec² θ = R.H.S. Proved.



More problems on trigonometric identities are shown where one side of the identity ends up with the other side.

4. . cos θ/(1 - tan θ) + sin θ/(1 - cot θ) = sin θ + cos θ

Solution:

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

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

= cos θ/{(cos θ - sin θ)/cos θ} + sin θ/{(sin θ - cos θ/sin θ)}

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

= (cos² θ - sin² θ)/(cos θ - sin θ)

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

= (cos θ + sin θ) = R.H.S. Proved.


5. Show that, 1/(csc A - cot A) - 1/sin A = 1/sin A - 1/(csc A + cot A)

Solution:

We have,

1/(csc A - cot A) + 1/(csc A + cot A)

= (csc A + cot A + csc A - cot A)/(csc² A - cot² A)

= (2 csc A)/1; [since, csc² A = 1 + cot² A ⇒ csc²A - cot² A = 1]

= 2/sin A; [since, csc A = 1/sin A]

Therefore,

1/(csc A - cot A) + 1/(csc A + cot A) = 2/sin A

⇒ 1/(csc A - cot A) + 1/(csc A + cot A) = 1/sin A + 1/sin A

Therefore, 1/(csc A - cot A) - 1/sin A = 1/sin A - 1/(csc A + cot A) Proved.


6. (tan θ + sec θ - 1)/(tan θ - sec θ + 1) = (1 + sin θ)/cos θ

Solution:

L.H.S = (tan θ + sec θ - 1)/(tan θ - sec θ + 1)

= [(tan θ + sec θ) - (sec² θ - tan² θ)]/(tan θ - sec θ + 1), [Since, sec² θ - tan² θ = 1]

= {(tan θ + sec θ) - (sec θ + tan θ) (sec θ - tan θ)}/(tan θ - sec θ + 1)

= {(tan θ + sec θ) (1 - sec θ + tan θ)}/(tan θ - sec θ + 1)

= {(tan θ + sec θ) (tan θ - sec θ + 1)}/(tan θ - sec θ + 1)

= tan θ + sec θ

= (sin θ/cos θ) + (1/cos θ)

= (sin θ + 1)/cos θ

= (1 + sin θ)/cos 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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