Using λ

Solutions shown graphically:

Plot[{Tan[Sqrt[λ]],-Sqrt[λ]},{λ,0,10Pi}]

Solve does not work here but we can use FindRoot to get numerical solutions one at a time

Solve[Tan[Sqrt[λ]]==-Sqrt[λ],λ]

First eigenvalue and eigenfunction:

FindRoot[Tan[Sqrt[λ]]==-Sqrt[λ],{λ,4}]

λ[1]=λ/.%

λ[1]

Plot[Sin[Sqrt[λ[1]]x],{x,0,1}]

Second eigenvalue and eigenfunction:

FindRoot[Tan[Sqrt[λ]]==-Sqrt[λ],{λ,24}]

λ[2]=λ/.%

Plot[Sin[Sqrt[λ[2]]x],{x,0,1}]

Using α

Determining equation

Solutions shown graphically:

Plot[{Tan[α],-α},{α,0,5Pi},Exclusions->{Pi/2},ExclusionsStyle->Red]

Plot[{Tan[α],-α},{α,0,5Pi},Exclusions->Table[(2k-1)Pi/2,{k,1,10}],ExclusionsStyle->Gray]

Solve does not work here but we can use FindRoot to get numerical solutions one at a time

Solve[Tan[α]==-α,α]

First eigenvalue and eigenfunction:

FindRoot[Tan[α]==-α,{α,2}]

α[1]=α/.%

Plot[Sin[α[1]x],{x,0,1}]

Second eigenvalue and eigenfunction:

FindRoot[Tan[α]==-α,{α,5}]

α[2]=α/.%

Plot[Sin[α[2]x],{x,0,1}]

Third eigenvalue and eigenfunction:

FindRoot[Tan[α]==-α,{α,8}]

α[3]=α/.%

Plot[Sin[α[3]x],{x,0,1}]

Expanding the function f(x)=1 in this orthogonal set

γ[1]=Integrate[1*Sin[α[1]x],{x,0,1}]/Integrate[Sin[α[1]x]^2,{x,0,1}]

γ[2]=Integrate[1*Sin[α[2]x],{x,0,1}]/Integrate[Sin[α[2]x]^2,{x,0,1}]

γ[3]=Integrate[1*Sin[α[3]x],{x,0,1}]/Integrate[Sin[α[3]x]^2,{x,0,1}]

Plot[{1,γ[1]Sin[α[1]x]+γ[2]Sin[α[2]x]+γ[3]Sin[α[3]x]},{x,0,1},PlotRange->{0,1.5}]