Question

if a + b is equal to 45 degrees then the value of 1 + tan ainto 1 + tan​

Answer 1

Given

$\sf a+b=45\degree$

To find

$\sf The \:value \:of\\ \sf (1+tan a)(1+tan b)$

Explanation

Consider

$a+b=45\degree$

Applying tan on both sides

$tan(a+b)=tan 45\degree$

$\underline{tan45\degree=1}$

then

$\dfrac{tana + tanb}{1 - tanatanb} = 1$

$tana + tanb = 1 - tanatanb$

$tana + tanb + tanatanb = 1.........(1)$

Now consider

$(1 + tana)(1 + tanb)$

Expanding

$1 + tanb + tana + tanatanb$

$= > 1 + 1 \: \: \: \: \: (from \: (1))$

$= > 2$

${\boxed{ \underline{(1 + tana)(1 + tanb) = 2}}}$

Formula used :

$\bullet tan(A+B)=\dfrac{tanA+tanB}{1-tanAtanB}$