Trigonometric Identity Involving Cosecant And Cotangent


 
 
Concept Explanation
 

Trigonometric Identity Involving Cosecant And Cotangent

Trigonometric Identities: The trigonometric identities are equalities which are true for every value appearing on both sides of the equal sign An equation involving trigonometric ratios of an angle is called a trigonometric identity, if it is true for all values of the angle(s) involved. These identities are useful whenever expressions involving trigonometric ratios are to be simplified.

Trigonometric Identity Involving Cesecant And Cotangent: The identity is

cosec^2;theta - cot^2;theta = 1

Let us try to prove these identities and use it further to simplify the various trigonometric expressions

In large Delta ABC, right -angled at B according to the pythagoras theorem  we have

                      AB^{2}+BC^{2}=AC^{2}                                  (1)

Proof For The Identity: Let us see what we get on dividing (1) by BC^{2}. We get

          frac{AB^{2}}{BC^{2}}+frac{BC^{2}}{BC^{2}}=frac{AC^{2}}{BC^{2}}

i.e.,    left (frac{AB}{BC} right)^{2}+left (frac{BC}{BC} right)^{2}= left (frac{AC}{BC} right)^{2}

i.e.,        cot^{2}A+1=cosec^{2}A  

             cosec^{2}A -cot^{2}A=1

This is true for all A such that 0^{circ}leq Aleq 90^{circ}. So, this is a trigonometric identity..

Illustration: Simplify the expression:

frac{sin Theta -2 sin^3Theta }{2 cos^3Theta -cos Theta }

Solution: We will simplify the expression using trigonometric Identities

frac{sin Theta -2 sin^3Theta }{2 cos ^3Theta -cos Theta }   =   frac{sin Theta (1-2 sin^2Theta )}{cos Theta (2 cos ^2Theta -1)}

                                  =tan;Theta left (frac{1-2(1-cos^2Theta )}{2 cos^2Theta -1} right)

                                 =tan;Theta left (frac{1-2 +2cos^2Theta )}{2 cos^2Theta -1} right)

                                 =tan Theta left (frac{2 cos^2Theta -1}{2 cos^2Theta -1} right)

                                 =tan Theta

Question : - Given that cosec Theta - cot Theta = 5, find the value of cosec Theta + cot Theta and sin Theta.

Sol. - We have, cosec^2 ; Theta - cot^2 Theta = 1

Rightarrow ;;;;; (cosecTheta-cotTheta)(cosec Theta + cot Theta)=1

Rightarrow ;;;;; 5(cosecTheta+cotTheta)=1 ;;Rightarrow ;; cosecTheta+cot Theta=frac {1}{5}                                      ............(i)

Now, cosec Theta - cot Theta = 5                                                                                                                 ...........(ii)

Add (i) and (ii), we get

2; cosec ; Theta = 5 + frac {1}{5} =frac {26}{5}

Rightarrow ;;;;; cosec Theta=frac {13}{5} ;; Rightarrow ;; sin Theta = frac {5}{13}

                   

Sample Questions
(More Questions for each concept available in Login)
Question : 1

Simplify the given expression :  (cosectheta-cottheta)^2 

Right Option : A
View Explanation
Explanation
Question : 2

cosec^2;theta - cot^2;theta =

Right Option : A
View Explanation
Explanation
Question : 3

If     tantheta=tinyfrac{1}{sqrt7}  , find the value of   frac{cosec^2theta-sec^2theta}{cosec^2theta+sec^2theta}

Right Option : A
View Explanation
Explanation
 
 


Students / Parents Reviews [10]