Show that: 2tan-1{tan(α/2) tan(( /4)-(β/2)) = tan-1{(sinαcosβ)/(cosα+sinβ)} - Maths - Inverse Trigonometric Functions

Question

Show that:
2tan-1{tan(α/2) tan((π/4)-(β/2)) =
tan-1{(sinαcosβ)/(cosα+sinβ)}

Ajanta Trivedi · Accepted Answer

if
α, β∈0,π2
since 0<α<π20<α2<π4⇒0<tanα2<1 
(1)since , 0<β<π20<β2<π4⇒-π4<-β2<0⇒0<π4-β2<π4⇒0<tan(π4-β2)<1  
(2)
from eq(1) and eq(2), we get:
0<tanα2 tan(π4-β2)<1
since, 2 tan-1x=tan-12x1-x2 , where x∈(0,1)
LHS of the given equation is:
2tan-1tanα2
tanπ4-β2=tan-12tan(α/2)tan(π/4-β/2)1-tan2α/2
tan2(π/4-β/2)=tan-12tanα2
1-tanβ/21+tanβ/21-tan2α/2
1-tanβ/21+tanβ/22=tan-12 tanα/2
(1-tanβ/2)(1+tanβ/2)(1+tanβ/2)2-tan2α/2
(1-tanβ/2)2=tan-12tanα/2
1-tan2β/21+tan2β/2+2tanβ/2-tan2α/2
(1+tan2β/2-2tanβ/2)
=tan-12tanα/2
(1-tan2β/2)(1+tan2β/2)(1-tan2α/2)+2tanβ/2(1+tan2α/2)=tan-12tanα/21+tan2α/2
1-tan2β/21+tan2β/21-tan2α/21+tan2α/2+2tanβ/21+tan2β/2=tan-1sinα
cosβcosα+cosβ

hope this helps you

Show that: 2tan-1{tan(α/2).tan((π/4)-(β/2)) = tan-1{(sinαcosβ)/(cosα+sinβ)}

Show that:
2tan^-1{tan(α/2).tan((π/4)-(β/2)) = tan^-1{(sinαcosβ)/(cosα+sinβ)}