Question

Please how I can prove that ? Cos^2(t)=1/1+tan^2(t) Thanks

Answer 1

I think you mean "prove" not "improve". See below

Explanation:

Consider the RHS

`1/( 1+ tan^2(t))`

`tan(t) = sin(t)/cos(t)`

So, `tan^2(t) = sin^2(t)/cos^2(t)`

So RHS is now:
`1/( 1+ (sin^2(t)/cos^2(t))`

`1/(( cos^2(t)+ sin^2(t))/cos^2(t))`

`cos^2(t)/( cos^2(t)+ sin^2(t))`

Now : `cos^2(t)+ sin^2(t) =1`

RHS is `cos^2(t)`, same as LHS.

QED.

Answer 2

`"see explanation"`

Explanation:

`"to prove this is an identity either manipulate the left side"`
`"into the form of the right side or manipulate the right side"`
`"into the form of the left side"`

`"using the "color(blue)"trigonometric identities"`

`•color(white)(x)tanx=sinx/cosx" and "sin^2x+cos^2x=1`

`"consider the right side"`

`rArr1/(1+sin^2t/cos^2t)`

`=1/((cos^2t+sin^2t)/cos^2t)`

`=1/(1/cos^2t)`

`=1xxcos^2t/1=cos^2t=" left side hence proved"`