If cosecA-sinA=x^3 and secA-cosA=y^3, then show that x^2y^2{x^2+y^2}=1

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hala718 | High School Teacher | (Level 1) Educator Emeritus

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`csecA - sinA= x^3 `

`==> 1/sinA - sinA = x^3`

`==> (1-sin^2 A)/sinA = x^3`

`==> (cos^2 A)/sinA = x^3 `

`==> x= ((cos^2A)/sinA)^(1/3)...............(1)`

`secA - cosA = y^3`

`==> 1/cosA - cosA = y^3`

`==> (1-cos^2 A)/cosA = y^3`

`==> (sin^2 A)/cosA = y^3`

`==> y= ((sin^2 A)/cosA)^(1/3)......................(2)`

Now we will substitute into the equations:

`(x^2 y^2)(x^2+y^2) = 1`

`((cos^2 A)/sinA)^(2/3) ((sin^2 A)/cosA)^(2/3) (((cos^2 A)/sinA)^(2/3) + ((sin^2 A)/cosA)^(2/3)) = 1`

`((cosA sinA)^(4/3) )/ (sinA cosA)^(2/3) (((cos^(4/3) A)(cos^ (2/3) A)+(sin^(4/3) A) (sin^(2/3) A))/(sinA cosA)^(2/3))= 1`

`==> (cosA sinA)^(2/3)((( cos^2 A) + (sin^2 A))/ (sinA cosA)^(2/3))= 1`

Reduce similar terms.

==> `cos^2 A + sin^2 A = 1`

`==> 1 = 1`

``Then, we have proved that if  `secA -cosx = y^3 `   and  `csecA-sinA= x^3 `  Then, `(x^2 y^2) (x^2 +y^2) = 1`

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