Question

Prove that: (1 + cos A)^2 - (1 - cos A)^2/
sin^2 A=
4cosec A. cotA​

Answer 1

see explanation

Explanation:

Using the trigonometric identities

cosec x =
`(1)/(sinx)` , cot x =
`(cosx)/(sinx)`

Consider the left side

`((1+cosA)^2-(1-cosA)^2)/(sin^2A)` ← expand and simplify numerator

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

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

=
`(4cosA)/(sin^2A)`

= 4 ×
`(1)/(sinA)` ×
`(cosA)/(sinA)`

= 4cosecAcotA

= right side, thus proven