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Trigonometric Identity Involving Cosecant And Cotangent

Introduction to Trigonometry II - Concepts

Trigonometric ratios of Complementary Angles

Trigonometric Identity Involving Sine And Cosine

Trigonometric Identity Involving Secant And Tangent

Trigonometric Identity Involving Cosecant And Cotangent

Trigonometric Identities

Class - 10th IMO Subjects

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 : $(cosectheta-cottheta)^2$
A. $frac{1-costheta}{1+costheta}$	B. $frac{1+costheta}{1-costheta}$	C. $frac{costheta}{1+costheta}$	D. $frac{1-costheta}{costheta}$
Right Option : A
View Explanation

Question : 2

If $tantheta=tinyfrac{1}{sqrt7}$ , find the value of $frac{cosec^2theta-sec^2theta}{cosec^2theta+sec^2theta}$
A. $frac{3}{4}$	B. $frac{4}{3}$	C. $frac{7}{8}$	D. $frac{8}{7}$
Right Option : A
View Explanation