Trigonometry is an important part of mathematics. It involves its own functions sine, cosine, etc. They are essential tools for simplifying expressions and solving trigonometric equations.

In this article, we will learn about trigonometric identities, their important formulas, and how to solve related equations.

What are Trigonometric Identities?

Trigonometric identities are mathematical equations involving trigonometric functions (like sine, cosine, and tangent) that hold for all values of the variables within their domains. They are essential tools for simplifying expressions and solving trigonometric equations.

Important Formulas

Various important trigonometric identities are:

• sin² θ + cos² θ = 1
• tan² θ = sec² θ – 1
• cosec²θ = 1+ cot²θ

• sin (-θ) = – Sin θ
• cos (-θ) = Cos θ
• tan (-θ) = – Tan θ
• cot (-θ) = – Cot θ
• sec (-θ) = Sec θ
• cosec (-θ) = -Cosec θ

• sin(x+y) = sin(x)cos(y)+cos(x)sin(y)
• cos(x+y) = cos(x)cos(y)–sin(x)sin(y)
• sin(x–y) = sin(x)cos(y)–cos(x)sin(y)
• cos(x–y) = cos(x)cos(y) + sin(x)sin(y)

• cos(2x) = 2cos²(x)-1 = 1–2sin²(x)= cos²(x)-sin²(x)
• Sin 3x = 3sin x – 4sin³x
• Cos 3x = 4cos³x-3cos x

Inverse Trigonometry Formulas

• sin^-1 (–x) = – sin^-1 x
• cos^-1 (–x) = π – cos^-1 x
• tan^-1 (–x) = – tan^-1 x
• cosec^-1 (–x) = – cosec^-1 x
• sec^-1 (–x) = π – sec^-1 x
• cot^-1 (–x) = π – cot^-1 x

Trigonometric Identities Practice Questions

Question 1: Prove that (sin⁴θ – cos⁴θ +1) cosec²θ = 2

Solution:

L.H.S. = (sin⁴θ – cos⁴θ +1) cosec²θ

= [(sin²θ – cos²θ) (sin²θ + cos²θ) + 1] cosec²θ

Using the identity sin²A + cos²A = 1,

= (sin²θ – cos²θ + 1) cosec²θ

= [sin²θ – (1 – sin²θ) + 1] cosec²θ

= 2 sin²θ cosec²θ

= 2 sin2θ (1/sin²θ)

= 2

= RHS

Question 2: Solve expression: cosec²θ – cot²θ.

Solution:

1 + cot²θ = cosec²θ

cosec²θ-cot²θ = 1+cot²θ-cot²θ

= 1 = R.H.S.

Question 3: (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.

Question 4: 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.

Question 5: (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

Question 6: (1 + cos θ)/sin θ + sin θ/(1 + cos θ) = 2 cosec θ

Solution:

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

= (1 + cos θ)²/[sin θ(1 + cos θ)] + sin θ²/[sin θ(1 + cos θ)]

= [(1 + cos θ)² + sin θ²]/[sin θ(1 + cos θ)]

= [1 + 2 cos θ + cos² θ + sin² θ]/[sin θ(1 + cos θ)]

= [1 + 2 cos θ + 1]/[sin θ(1 + cos θ)]; [since cos² θ + sin² θ = 1]

= [2 + 2 cos θ]/[sin θ(1 + cos θ)]

= 2(1 + cos θ)/[sin θ(1 + cos θ)]

= 2/sin θ

= 2 cosec θ = R.H.S. (Proved)

Question 7: Prove cosθ/(1+sinθ) = (1-sinθ)/cosθ.

Solution:

cosθ/(1+sinθ){(1-sinθ)/(1-sinθ)} {multiplying and dividing by 1-sinθ on LHS}

= cosθ(1-sinθ)/1-sin²θ

= cosθ(1-sinθ)/cos²θ

= 1-sinθ/cosθ = R.H.S.

Worksheet: Trigonometric Identities

Problem 1: Prove it sin² θ + cos² θ = 1.

Problem 2: Prove it tan² θ = sec² θ – 1.

Problem 3: Prove it sin(x+y) = sin(x)cos(y)+cos(x)sin(y).

Problem 4: Prove it cos(x+y) = cos(x)cos(y)–sin(x)sin(y).

Problem 5: Prove that (cos A – sin A + 1)/ (cos A + sin A – 1) = cosec A + cot A, using the identity cosec²A = 1 + cot²A.

Problem 6: Prove it sin(x–y) = sin(x)cos(y)–cos(x)sin(y).

Problem 7: Prove it cos(x–y) = cos(x)cos(y) + sin(x)sin(y).

Problem 8: If a cos³α + 3a cos α sin²α = m and a sin³α + 3a cos²α sin α = n, then find (m + n)^2/3 + (m – n)^2/3.

Problem 9: Prove it cos(2x) = 2cos²(x)-1 = 1–2sin²(x).

Problem 10: Prove it Sin 3x = 3sin x – 4sin³x.

Problem 11: Prove it Cos 3x = 4cos³x-3cos x.

Problem 12: Prove that: (cosec A – sin A)(sec A – cos A) = 1/(tan A + cot A).

Also, Read:

• List of All Trigonometric Identities
• Important Trigonometry Question
• Inverse Trigonometric Identities

Frequently Asked Questions

What are Trigonometric Identities?

Trigonometric identities are mathematical equations involving trigonometric functions (like sine, cosine, and tangent) that hold true for all values of the variables within their domains. They are essential tools for simplifying expressions and solving trigonometric equations.

How are Trigonometric Identities Used in Solving Equations?

Trigonometric Identities can transform complex trigonometric equations into simpler forms. By applying these identities, you can combine or break down trigonometric terms, making it easier to find solutions to equations involving angles.

What is Importance of Trigonometric Identities?

Trigonometric identities simplify complex trigonometric expressions, making it easier to solve equations, integrate functions, and analyze waveforms in various applications such as physics, engineering, and computer graphics.

How Do Trigonometric Identities Apply to Real-World Problems?

Trigonometric identities are used in signal processing, electrical engineering, navigation, and physics to model periodic phenomena such as sound waves, light waves, and alternating current circuits.