If one of the trigonometric ratios of an acute angle is known, the remaining trigonometric ration of the angle can be determined.

Two angles are said to be complementary, if their sum is 90⁰ and each one of them is called the complement of the other.

Trigonometric Identities

An equation with trigonometric ratios of an angle θ, which is true for all values of ‘ θ ‘, for which the given trigonometric ratios are defined, is called an identity.

The three fundamental trigonometric identities are

1. sin² θ +cos² θ = 1

 ⇒ sin² θ =1-cos² θ

⇒ sin² θ =(1-cos θ)(1+cos θ)

⇒ (1- cos θ) = (sin² θ) /(1+ cos θ)

⇒ (1+ cos θ) = (sin² θ) /(1- cos θ)

⇒ cos² θ + sin² θ = 1

cos² θ =1- sin² θ

⇒ cos² θ =(1- sin θ)(1+ sin θ)

⇒ (1+ sin θ) = (cos² θ) /(1- sin θ)

⇒ (1- sin θ) = cos² θ /(1+sin θ)

(b) sec² θ = 1 + tan² θ

⇒ sec² θ – tan² θ =1

⇒ (sec θ – tan θ)(sec θ + tan θ) = 1

⇒ (sec θ – tan θ) = 1/ (sec θ + tan θ)

⇒ (sec θ + tan θ) = 1/ (sec θ – tan θ)

⇒ sec² θ – 1 = tan² θ

⇒ (sec θ – 1)( sec θ – 1) = tan² θ

(c) cosec² θ = 1+cot²θ

⇒ cosec² θ – cot² θ = 1

⇒ (cosec θ – cot θ)(cosec θ + cot θ)=1

(cosec θ+ cot θ) =1cosec θ – cot θ

(cosec θ- cot θ) = 1cosec θ + cot θ

⇒ Cosec² θ – 1 = cot² θ

⇒ (Cosec θ – 1)( Cosec θ – 1) = Cot² θ

Supportive Formulae:

(a+b)²=+a²+b²+2ab

(a-b)² = a²+b²-2ab

(a+b)²+(a-b)²= 2 (a²+b²)

(a+b)²– (a-b)²= 4ab

(a-b)²– (a+b)²= – 4ab

(a+b)² = (a-b)²+ 4ab

(a-b)² = (a+b)²– 4ab

(a²-b²)=(a+b)(a-b)

a+b=(a²-b²) /(a-b)

a-b=(a²-b²) /(a-b)

(a+b)²= (a-b)²+ 4ab