Question

How do we know that Tan(180-x)=-tanx And Cos(180-x)=-cosx Sin(180-x)=Sinx Tan(90-x)=1/-tanx etc...is there any way we can find out these?

Answer 1

See below

Explanation:

Let's try proving two them

`tan(180^@-x)=-tanx`

Apply tangent sum identity:
`(tan180^@-tanx)/(1+tan180^@tanx)`

Simplify:
`(0-tanx)/(1+0*tanx)`

`-tanx/1= -tanx`

The second one we will try is
`tan(90^@+x)=1/-tanx`:

Unfortunately `tan(90^@)` is undefined so:
Apply quotient identity:
`sin(90^@+x)/cos(90^@+x)=1/-tanx`:

Now apply the sine and cosine sum identities:
`(sin90^@cosx+cos90^@sinx)/(cos90^@cosx-sin90^@sinx)=1/-tanx`

Simplify:
`((1)cosx+(0)sinx)/((0)cosx-(1)sinx)=1/-tanx`

`cosx/-sinx=-1/tanx`

Apply quotient identity:
`-cotx=-1/tanx`

Apply reciprocal identity:
`-1/tanx=-1/tanx`