TRIGONOMETRY - Pythagorean Identities

Pythagorean Identities in Trigonometry :

sin2θ + cos2θ = 1

sin2θ = 1 - cos2θ

cos2θ = 1 - sin2θ

sec2θ - tan2θ = 1

sec2θ = 1 + tan2θ

tan2θ = sec2θ - 1

csc2θ - cot2θ = 1

csc2θ = 1 + cot2θ

cot2θ = csc2θ - 1

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)2sec2θ + 2tan θ

Solution :

= (1 + tan θ)2

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

= 12 + 2(1)tan θ + tan2θ

+ 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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