Prove that :( 1 + 1/tan^(2)A)(1 + 1/cot^(2)A) = 1/sin^(2)A - sin^(4)A

Question

Prove that :( 1 + 1/
`tan^(2)A`)(1 + 1/
`cot^(2)A`) = 1/
`sin^(2)A` -
`sin^(4)A`

Answer 1

See explanation

Explanation:

Simplify left and right parts separately.

Left part:

`\left(1+(1)/(\tan^2A)\right)\left(1+(1)/(\cot ^2A)\right)\\ \\=\left(1+(1)/((\sin^2A)/(\cos^2A))\right)\left(1+(1)/((\cos^2A)/(\sin^2A))\right)\\ \\=\left(1+(\cos^2A)/(\sin^2A)\right)\left(1+(\sin^2A)/(\cos^2A)\right)\\ \\=(\sin^2A+\cos^2A)/(\sin^2A)\cdot (\cos^2A+\sin^A)/(\cos^2A)\\ \\=(1)/(\sin^2A)\cdot (1)/(\cos^2A)\\ \\=(1)/(\sin^2A\cos^2A)`

Right part:

`(1)/(\sin^2A-\sin^4A)\\ \\=(1)/(\sin^2A(1-\sin^2A))\\ \\=(1)/(\sin^2A\cos^2A)`

Since simplified left and right parts are the same, then the equality is true.