TRIGONOMETRIC IDENTITIES EXAMPLES

Abbreviations used :

L.H.S -----> Left hand side

R.H.S -----> Right hand side

Example 1 : 

Prove :

tanθ/(1 - tan2θ)  =  sinθsin(90 - θ)/[2sin2(90 - θ) - 1]

Solution :

L.H.S

   =  tan θ/(1 - tan2θ)

  =  sinθsin(90 - θ)/[2sin2(90 - θ) - 1]

=  R.H.S

Example 2 : 

Prove :

[1/(cosecθ - cotθ)] - (1/sinθ) 

=  [(1/sinθ)] - [1/(cosecθ + cotθ)]

Solution :

L.H.S :

=  [1/(cosecθ - cotθ)] - (1/sinθ)

Multiply numerator and denominator of the first fraction by (cosecθ + cotθ).

=  [(1/(cosecθ - cotθ)) x (cosecθ + cotθ)/(cosecθ + cotθ)] - (1/sinθ)

=  (cosecθ  + cotθ)/(cosec2θ - cot2θ) - (1/sinθ)

=  [(cosecθ + cotθ)/1] - (1/sinθ)

=  [(cosecθ + cotθ) - cosecθ]

=  cotθ -----(1)

R.H.S :

=  (1/sinθ) - [1/(cosecθ - cotθ)]

Multiply the numerator and denominator of the second fraction by (cosecθ + cotθ).

 [(cosecθ - cotθ)/(cosecθ - cotθ)(cosecθ + cotθ)] - (1/sinθ)

=  (cosecθ + cotθ)/(cosec2θ - cot2θ) - (1/sinθ)

   =  [(cosecθ + cotθ)/1] - (1/sinθ)

   =  [(cosecθ + cotθ) - cosecθ]

   =  cotθ -----(2)

From (1) and (2), we get

L.H.S  =  R.H.S

Example 3 : 

Prove :

(cot2θ + sec2θ) / (tan2θ +  cosec2θ) 

=  sinθcosθ(tanθ + cotθ)

Solution :

L.H.S :

=  (cot2θ + sec2θ)/(tan2θ + cosec2θ)

=  (cosec2θ - 1 + 1 + tan2θ)/(tan2θ + cosec2θ)

=  (tan2θ + cosec2θ)/(tan2θ + cosec2θ)

=  1 -----(1)

R.H.S :

=  sinθcosθ(tanθ + cotθ)

=  sinθcosθ[(sinθ/cosθ) + (cosθ/sinθ)]

=  sinθcosθ[(sin2θ + cos2θ)/(cosθsinθ)]

  =  sinθcosθ [1/(cosθsinθ)]

=  1 -----(2)

From (1) and (2), we get

L.H.S  =  R.H.S

Example 4 :

If x = asecθ + btanθ and y = atanθ + bsecθ then prove that

x2 - y2  =  a2 - b2

Solution :

x2  =  (asecθ + btanθ)2

x2  =  (asecθ)2 + (btanθ)2 + 2absecθtanθ

x2  =  a2sec2θ + b2tan2θ + 2absecθtanθ -----(1)

y2  =  (atanθ + bsecθ)2

y2  =  (atanθ)2 + (bsecθ)2 + 2abtanθsecθ

y2  =  a2tan2θ + b2sec2θ + 2abtanθsecθ -----(2)

(1) - (2) :

x- y2  =  (a2sec2θ + b2tan2θ + 2absecθtanθ)

- (a2tan2θ + b2sec2θ + 2abtanθsecθ)

x- y2  =  a2sec2θ + b2tan2θ + 2absecθtanθ

- a2tan2θ - b2sec2θ - 2abtanθsecθ

x- y2  =  a2sec2θ + b2tan2θ a2tan2θ - b2sec2θ

x- y2  =  a2sec2θ - a2tan2θ - b2sec2θ + b2tan2θ

x- y2  =  a2(sec2θ - tan2θ) - b2(sec2θ - tan2θ)

x- y2  =  a2(1) - b2(1)

x- y2  =  a2 - b2

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.

©All rights reserved. onlinemath4all.com

Recent Articles

  1. Quadratic Equation Word Problems Worksheet with Answers

    Oct 07, 22 12:25 PM

    Quadratic Equation Word Problems Worksheet with Answers

    Read More

  2. Problems on Quadratic Equations

    Oct 07, 22 12:20 PM

    Problems on Quadratic Equations

    Read More

  3. SAT Math Practice Worksheets

    Oct 07, 22 09:38 AM

    SAT Math Practice Worksheets - Topic wise worksheet with step by step explanation for each question

    Read More