Question

Find the values of $tan^{-1}(\tan\frac{3\pi}{4})$

Answer 1

$tan^{-1}(\tan\frac{3\pi}{4})$
We know that $\tan^{-1}(\tan x)=x \,if x\in(-\frac{\pi}{2},\frac{\pi}{2})$, which is the principal value branch of $\tan^{-1}x$.
Here, 3π/4∉(-π/2'π/2)
Now,tan-1(tan(3π/4)can be written as:
tan-1(tan(3π/4)=tan-1[-tan(-3π/4)]=tan-1[-tan(π-π/4)]
=tan-1[-tanπ/4]=tan-1[tan(-π/4)] where,-π/4∈(-π/2'π/2)
therefore tan-1(tan(3π/4)=tan-1(tan(-π/4)=-π/4