Question

Question #4f2b1

Answer 1

`[(sinx+cosx)^2-1]/cos^2x=2tanx`

Explanation:

`[(sinx+cosx)^2-1]/cos^2x=((sin^2x+2sinxcosx+cos^2x)-1)/cos^2x`

Use the identity:
`sin^2x+cos^2x=1`

`[(sinx+cosx)^2-1]/cos^2x=(2sinxcosx+1-1)/cos^2x`

`[(sinx+cosx)^2-1]/cos^2x=(2sinxcosx)/cos^2x`

`[(sinx+cosx)^2-1]/cos^2x=(2sinx)/cosx`

Use the identity:
`sinx/cosx=tanx`

`[(sinx+cosx)^2-1]/cos^2x=2tanx`

Answer 2

`2tanx`

Explanation:

`"expand " (sinx+cosx)^2`

`rArr(sin^2x+2sinxcosx+cos^2x-1)/cos^2x`

`•color(white)(x)sin^2x+cos^2x=1`

`rArr(2sinxcosx+1-1)/cos^2x`

`=(2sinxcancel(cosx))/(cancel(cosx)cosx)=2sinx/cosx=2tanx`