WORKSHEET ON PROVING TRIGONOMETRIC IDENTITIES

Prove each of the following.

Problem 1 :

(1 - cos²θ)(1 + cot²θ) = 1

Problem 2 :

sin²θ + tan²θ = sec²θ - cos²θ 

Problem 3 :

secθ - tanθsinθ = cosθ

Problem 4 :

sin⁴θ + cos⁴θ = 1 - 2sin²θcos²θ

Problem 5 :

Problem 6 :

Problem 7 :

sin⁴θ - cos⁴θ = sin²θ - cos²θ

Problem 8 :

Problem 9 :

(secθ + cosθ)(secθ - cosθ) = tan²θ + sin²θ

Problem 10 :

Answers

1. Answer :

(1 - cos²θ)(1 + cot²θ) :

= (sin²θ)(csc²θ)

2. Answer :

sin²θ + tan²θ :

= 1 - cos²θ + sec²θ - 1

= -cos²θ + sec²θ

= sec²θ - cos²θ

3. Answer :

secθ - tanθsinθ :

= cosθ

4. Answer :

sin⁴θ + cos⁴θ :

= (sin²θ)² + (cos²θ)²

Using a² + b² = (a + b)² - 2ab,

= (sin²θ + cos²θ)² - 2sin²θcos²θ

= 1² - 2sin²θcos²θ

= 1 - 2sin²θcos²θ

5. Answer :

= secθ - tanθ

6. Answer :

7. Answer :

sin⁴θ - cos⁴θ :

= (sin²θ)² - (cos²θ)²

= (sin²θ + cos²θ)(sin²θ + cos²θ)

= (1)(sin²θ + cos²θ)

= sin²θ + cos²θ

8. Answer :

9. Answer :

(secθ + cosθ)(secθ - cosθ) :

= sec²θ - cos²θ

= 1 + tan²θ - cos²θ

= tan²θ + 1 - cos²θ

= tan²θ + sin²θ

10. Answer :

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