Table of Content

• What are Trigonometric Identities?
• What are Trigonometric Ratios?
• List of Trigonometric Identities Class 10
• Proof of Trigonometric Identities Class 10
• Solved Examples

Identities Name	Identities
Pythagorean Identities	sin2θ + cos2θ = 1
	1 + tan2θ = sec2θ
	1 + cot2θ = cosec2θ
Reciprocal Identities	cosec θ = 1/sin θ
	sec θ = 1/cos θ
	cot θ = 1/tan θ
Quotient Identities	tan θ = sin θ/cos θ
	cot θ = cos θ/sin θ

Also, Check

• Trigonometry Formulas
• Trigonometry Table
• Height and Distance

Solved Examples on Trigonometric Identities Class 10

Example 1: Prove that: (1 + cot²A) sin²A = 1

Solution:

LHS = (1 + cot²A) sin²A

Using trigonometric identity

1 + cot²θ = cosec²θ

⇒ LHS = cosec²A sin²A

⇒ LHS = (1 / sin²A) sin²A [cosec A = 1 / sin A]

⇒ LHS = 1

Thus, LHS = RHS

Hence Proved

Example 2: Prove that: tan²B cos²B = 1 – cos²B

Solution:

LHS = tan²B cos²B

Using trigonometric identity

⇒ 1 + tan²θ = sec²θ

⇒ tan²θ = sec²θ – 1

⇒ LHS = (sec²B – 1) cos²B

⇒ LHS = sec²Bcos²B – cos²B

⇒ LHS = [(1/ cos²B)cos²B] – cos²B

⇒ LHS = 1 – cos²B

Thus, LHS = RHS

Hence Proved

Example 3: Prove that: tan θ + (1/tan θ) = sec θ cosec θ

Solution:

LHS = tan θ + (1/tan θ)

⇒ LHS = (tan²θ + 1)/tan θ

Using trigonometric identity

1 + tan²θ = sec²θ

⇒ LHS = sec²θ/tanθ

⇒ LHS = (1 / cos²θ) × (cos θ / sinθ )

⇒ LHS = (1 / cos θ) × (1 / sin θ)

⇒ LHS = sec θ cosec θ [sec θ = 1/cos θ and cosec θ = 1/sin θ]

Thus, LHS = RHS

Hence Proved

Example 4: Prove that: √[(1 – cos θ)/(1 + cos θ)] = cosec θ – cot θ

Solution:

LHS = √[(1 – cos θ)/(1 + cos θ)]

⇒ LHS = √[{(1 – cos θ)/(1 + cos θ)} × {(1 – cos θ) / (1 – cos θ)}]

⇒ LHS = √[(1 – cos θ)²/(1 – cos²θ)]

Using trigonometric identity

sin²θ + cos²θ = 1

⇒ LHS = √[(1 – cosθ)² / sin²θ]

⇒ LHS = (1 – cos θ) / sin θ

⇒ LHS = (1 / sin θ) – (cos θ / sin θ)

⇒ LHS = cosec θ – cot θ [cotθ = cosθ / sinθ and cosecθ = 1/sinθ]

Thus, LHS = RHS

Hence Proved

Example 5: Prove that: sin θ / (1 – cos θ) = cosec θ + cot θ

Solution:

LHS = sin θ/(1 – cos θ)

⇒ LHS = [sin θ/(1 – cos θ)] × [(1 + cos θ)/(1 + cos θ)}]

⇒ LHS = [sin θ (1 + cos θ)] / (1 – cos²θ)

Using trigonometric identity

sin²θ + cos²θ = 1

⇒ LHS = [sin θ (1 + cos θ)] / sin²θ

⇒ LHS = (1 + cos θ)/sin θ

⇒ LHS = (1 / sin θ) + (cos θ/sinθ)

⇒ LHS = cosec θ + cot θ [cot θ = cos θ/sin θ and cosec θ = 1/sin θ]

Thus, LHS = RHS

Hence Proved

Example 6: Prove that: [(1 + cot²θ)tan θ]/sec²θ = cot θ

Solution:

LHS = [(1 + cot²θ)tan θ]/sec²θ

Using trigonometric identity

1 + cot²θ = cosec²θ

⇒ LHS = [cosec²θ tan θ]/sec²θ

⇒ LHS = (1/sin²θ) × (sin θ/cos θ) × (cos²θ/1)

⇒ LHS = (cos θ/ sin θ)

⇒ LHS = cot θ [cot θ = cos θ/sinθ]

Thus, LHS = RHS

Hence Proved

Practice Questions on Trigonometric Identities Class 10

Q1: Prove that: cosecθ√[(1 – cos²θ) = 1

Q2: Prove that: (sec²x – 1)(cosec²x – 1) = 1

Q3: Prove that: sin²C + [1 / (1 + tan²C)] = 1

Q4: Prove that: sec A (1 – sin A) (sec A + tan A) = 1

Q5: Prove that: (1 – sin θ) / (1 + sin θ) = (sec θ – tanθ)²

Q6: Prove that: (cosec θ + sin θ)(cosec θ – sin θ) = cot²θ + cos²θ

Trigonometric Identities Class 10 – FAQs

What are Trigonometric Identities?

The trigonometric identities are the identities which gives the connection between the trigonometric identities.

List the Three basic Trigonometric Identities in Class 10.

The three basic trigonometric identities are:

• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• 1 + cot²θ = cosec²θ

What are the Six Trigonometric Ratios?

The six basic trigonometric ratios are: sin, cos, tan, cosec, sec and cot.

How to Prove Trigonometric Identities Class 10?

To prove trigonometric identities class 10 we use Pythagoras theorem. The detailed proof has been discussed in the article above.

Is Trigonometric Identities Class 10 Important?

Yes, Trigonometric Identities Class 10 is extremely important for board exams as well as for building strong foundation in Trigonometry