Question # EXAMPLE 27 Prove that tan
-1
X

# EXAMPLE 27 Prove that tan
-1
X

Solution SOLUTION We have
[
mathrm{LHS}=tan ^{-1}left(frac{cos x-sin x}{cos x+sin x}right)
]
( =tan ^{-1}left(frac{1-tan x}{1+tan x}right)[text { dividing num. and denom. by } cos x] )
[
begin{aligned}
&=tan ^{-1}left{tan left(frac{pi}{4}-xright)right}=left(frac{pi}{4}-xright)=mathrm{RHS}
therefore tan ^{-1}left(frac{cos x-sin x}{cos x+sin x}right) &=left(frac{pi}{4}-xright)
end{aligned}
]

Solution