Trigonometric Identity Involving Secant And Tangent


 
 
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Trigonometric Identity Involving Secant And Tangent

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 Secant And Tangent:  The identity can be expressed as

sec^2;theta - tan^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 now divide (1) by large AB^{2}. We get

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

or,         left (frac{AB}{AB} right)^{2}+left (frac{BC}{AB} right)^{2}=left (frac{AC}{AB} right)^{2}

i.e.,       (1)^{2}+(tan ;A)^{2}= (sec;A)^2

i.e.,       sec^{2}A-tan^{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 Identites

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

We have, sec^2; A =1 ; + ; tan^2 ; A =1;+; frac {1}{cot^2; A} = frac {1 +cot^2 ; A}{cot^2;A}

   Rightarrow ;;;;; sec ; A =frac {sqrt {1+cot^2 ; A}}{cot ; A}

                   

Sample Questions
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Question : 1

1 + tan^2;theta =

Right Option : B
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