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?

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Answer 1 · 2018-04-20T23:54:00

See below

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:
$-1/tanx=-1/tanx$