Question

Does `2+2tanx=2secx`?

Answer 1

`2+2tanx!=2secx`

Explanation:

`2+2tanx=2secx ?`

`2+(2sinx)/cosx=2/(cosx)`

`(2(cosx))/cosx+(2sinx)/cosx=2/(cosx)`

`(2cosx+2sinx)/cosx=2/(cosx)`

`(2(cosx+sinx))/cosx=2/(cosx)`

For this to be an identity, `cosx+sinx` must equal 1 for all values of `x`. But, for instance, with `x=pi/2 => sinx=cosx=sqrt2/2`.

`sqrt2/2+sqrt2/2=sqrt2!=1`

And so the identity is not valid.

`2+2tanx!=2secx`

Answer 2

Only when `x=2nxx360^@`, where `n` is an integer.

Explanation:

No, if you mean `2+2tanx=2secx` as an identity it is not.

But we can solve this for `x` and at some values we may have

`2+2tanx=2secx`.

As `2+2tanx=2secx`

`1+tanx=secx`

and multiplying by `cosx` we get

`cosx+sinx=1`

or `sqrt2(cosx xx1/sqrt2+sinx xx1/sqrt2)=1`

or `cosx xx1/sqrt2+sinx xx1/sqrt2=1/sqrt2`

or `cosxcos45^@+sinxsin45^@=cos45^@`

or `cos(x-45^@)=cos45^@`

i.e. `x-45^@=2nxx360^@+-45^@`, where `n` is an integer.

i.e. `x=2nxx360^@=90^@` or `x=2nxx360^@`

But at first solution, both `tanx` and `secx` are not defined

Hence `x=2nxx360^@`

graph{(y-secx+tanx+1)=0 [-10, 10, -5, 5]}