TRIGONOMETRY - Pythagorean Identities

Pythagorean Identities in Trigonometry :

Solved Problems

Problems 1 - 5 : Prove the given statment.

Problem 1 :

Solution :

Problem 2 :

Solution :

Problem 3 :

Solution :

Problem 4 :

Solution :

Problem 5 :

Solution :

Problem 6 :

Solution :

Problem 7 :

(sec θ + tan θ)(sec θ + tan θ) = 1

Solution :

= (sec θ + tan θ)(sec θ + tan θ)

Using the algebraic identity a² - b² = (a + b)(a - b),

= sec²θ - tan²θ

= 1

Problem 8 :

(1 + tan θ)² = sec²θ + 2tan θ

Solution :

= (1 + tan θ)²

Using the algebraic identity (a² + b²) = a² + 2ab + b².

= 1² + 2(1)tan θ + tan²θ

= 1 + tan²θ + 2tan θ

= sec²θ + 2tan θ

Problem 9 :

sec²θ - csc²θ = tan²θ - cot²θ

Solution :

= sec²θ - csc²θ

= sec²θ - (1 + cot²θ)

= sec²θ - 1 - cot²θ

= tan²θ - cot²θ

Problem 10 :

sin⁴θ - cos⁴θ = sin²θ - cos²θ

Solution :

= sin⁴θ - cos⁴θ

= (sin²θ)² - (cos²θ)²

Using the algebraic identity a² - b² = (a + b)(a - b),

= (sin²θ + cos²θ)(sin²θ - cos²θ)

= (1)(sin²θ - cos²θ)

= sin²θ - cos²θ

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