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

Trigonometric Identities are the rules that are followed by the Trigonometric Ratios. Trigonometric Identities and ratios are the fundamentals of trigonometry. Trigonometry is used in various fields for different calculations. The trigonometric identities class 10 gives the connection between the different trigonometric ratios. This article covers trigonometric identities class 10 in addition to their proofs.

This article will be extremely helpful for class 10 students for their exams. Also, we will solve some examples of trigonometric identities Class 10. Let’s discuss the topic of Trigonometric Identities Class 10 in depth.