What is trigonometric identity? Explain with examples.

Solution:

An equation is called an identity when it is true for all values of the variables involved. Similarly, an equation involving trigonometric ratios of an angle is called a trigonometric identity, if it is true for all values of the angles involved.

In a right-angled triangle, the three different sides are perpendicular, hypotenuse and base.

There are a total of six trigonometric ratios. They are as follows:

sin θ  = perpendicular / hypotenuse

cos θ = base / hypotenuse

tan θ = perpendicular / base

cot θ = base / perpendicular 

cosec θ = hypotenuse / perpendicular

sec θ = hypotenuse /  base

From the given ratios, we observe that sin and cosec are reciprocal of each other. These are called reciprocal identities

Reciprocal Identities
sinθ = 1 / cosecθ  ; cosec θ = 1/sinθ
cosθ = 1/secθ ; secθ = 1/cosθ
tanθ = 1 / cotθ ; cotθ= 1/tanθ

Quotient Identities
tanθ = sinθ/cosθ
cotθ = cosθ/sinθ

We are deriving all the different identities of trigonometry with the help of the Pythagoras theorem.

Let us understand one of the identities through the Pythagoras theorem.

According to Pythagoras theorem,

(perpendicular)² + (base)² = (hypotenuse)²   -------- (1) 

Let us write sin A  = perpendicular / hypotenuse   ------ (2)

and cos A = base / hypotenuse   ------- (3)

On squaring and adding equation (2) and (3), we get

sin² A + cos² A =  (perpendicular / hypotenuse)² + (base / hypotenuse)²

sin² A + cos² A = (perpendicular ² + base²) / hypotenuse²

sin² A + cos² A =  hypotenuse²/  hypotenuse² [from equation (1) ]

sin² A + cos² A = 1 

Hence , sin² A + cos² A = 1 is an identity of trigonometry.

Similarly we can also derive other identities by using pythagoras theorem.

Pythagorean Identities
sin2θ + cos2θ = 1 (⇒ cos2θ = 1 - sin2θ; sin2θ = 1 - cos2θ)
1 + tan2θ = sec2θ (⇒ tan2θ = sec2θ - 1 ; sec2θ - tan2θ = 1)
cosec2θ - cot2θ = 1 ( ⇒ 1 + cot2θ = cosec2θ ; cot2θ = cosec2θ - 1)

Example: If sin θ is 4 / 5, then find the value for cos θ.

Solution: Using trigonometric identity

sin²θ + cos²θ = 1 

⇒ cos²θ = 1 - sin²θ 

⇒ cos²θ = 1 - (16/25)

⇒ cos²θ = 9 / 25

⇒ cos θ = 3 /5  [ taking square root of 9 / 25 ]

There are other identities like opposite angle identities, half-angle identities, cofunction identities  some of it are listed below

Opposite Angle Identities (Even Odd Identities)
cos(−θ) = cos(θ), sec(−θ) = sec(θ)
tan(−θ) = −tan(θ), cot(−θ) = - tan(θ)
sin(−θ) = −sin(θ), cosec(−θ) = - cosec(θ)

Double Angle Identities
sin2θ = 2 sinθcosθ  = 2 tanθ/ (1 + tan2θ)
cos2θ = cos2θ - sin2θ  (= (1- tan2θ)/ (1 + tan2θ)
tan2θ = 2tanθ/ 1- tan2θ

Cofunction Identities
sin(π/2 - θ) = cosθ ;  cos (π/2 - θ) = sinθ ;
tan(π/2 - θ) = cotθ ;  cot(π/2 - θ) = tanθ ;
sec(π/2 - θ) = cosecθ ;  cosec (π/2 - θ) = secθ ;

Thus, trigonometric ratios of sin, cos, tan cot, sec, cosec are connected through a set of rules or identities listed above.

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What is trigonometric identity? Explain with examples.

Summary:

Trigonometric ratios of sin, cos, tan, cot, sec, cosec are connected through a set of rules or identities.