Prove the trigonometric identity 1 /(cosec A - cotA) - 1/sinA = 1/sinA - 1/(cosecA + cotA) 

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embizze | High School Teacher | (Level 1) Educator Emeritus

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In a mathematical proof, and verifying identities is a form of proof, you cannot assume what you are trying to prove. (Logically, you can prove anything from a false premise.) It is better to work from one side, and show that you can get to the other side.

We will start from the left-hand side (LHS) and show that you can get to the RHS:

`LHS=1/(cscA-cotA)-1/sinA`                         Given

`=(sinA-(cscA-cotA))/(sinA(cscA-cotA))`                  Add the fractions

`=(sinA-1/sinA+cosA/sinA)/(sinA(1/sinA-cosA/sinA))`                Rewrite in terms of sin and cos

`=((sin^2A-1+cosA)/sinA)/(1-cosA)`               Add the fractions in the numerator

`=((cosA-cos^2A)/sinA)/(1-cosA)`                        Use the pythagorean identity

`=((cosA(1-cosA))/sinA)/(1-cosA)`

`=cosA/sinA`

`=((cosA(1+cosA))/sinA)/(1+cosA)`   ``

`=((cosA+cos^2A)/sinA)/(1+cosA)`                 Now use the pythagorean identity

`=((cosA+1-sin^2A)/sinA)/(1+cosA)`            

`=(1/sinA+cosA/sinA-sinA)/(1+cosA)`

`=(cscA+cotA-sinA)/(sinA(1/sinA+cosA/sinA))`

`=(cscA+cotA-sinA)/(sinA(cscA+cotA))`

`=1/sinA-1/(cscA+cotA)`

`=RHS` as required.

Sources:
justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

The identity `1/(cosec A - cot A) - 1/sinA = 1/sinA- 1/(cosec A + cot A)` has to be proved.

If `1/(cosec A - cot A) - 1/sinA = 1/sinA- 1/(cosec A + cot A)`

=> `1/(cosec A - cot A) + 1/(cosec A + cot A) = 2/sin A`

`1/(cosec A - cot A) + 1/(cosec A + cot A)`

=> `(cosec A + cot A + cosec A - cot A)/((cosec A - cot A)(cosec A + cot A)`

=> `(2*cosec A)/(cosec^2 A - cot^2A)`

=> `(2*cosec A)/(1/(sin^2A) - (cos^2A)/(sin^2A))`

=> `(2*cosec A)/((1 - cos^2A)/(sin^2A))`

=> `(2*cosec A)/((sin^2A)/(sin^2A))`

=> `2*cosec A`

=> `2/sin A`

This proves `that 1/(cosec A - cot A) - 1/sinA = 1/sinA- 1/(cosec A + cot A)`

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lochana2500 | Student, Undergraduate | (Level 1) Valedictorian

Posted on

I found an easy way to prove that

L:HS ≡ 1/(cosecA - cotA) - cosecA

= {(cosecA + cotA)/cosec²A - cot²} - cosecA

we know that cosec²A - cot²A = 1

= cosecA + cotA - cosecA

= cosecA - (cosecA - cotA)

= cosecA - (cosecA - cotA)(cosecA + cotA)/(cosecA + cotA)

= cosecA -(cosec²A - cot²A)/(cosecA + cotA)

= 1/sinA - 1/(cosecA + cotA)  

= R:H:S

This proves the trigonometric identity..

Thank for helped me....

 

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