Find the exact value algebraically tan105 SolutionWe have gi

Find the exact value algebraically. tan(-105°)

Solution

We have given tan(-105⁰)

tan(-105⁰)=-tan(105⁰)

=-tan(60⁰+45⁰)

we use tan(A+B)=[tanA+tanB]/(1-tanA*tanB) formula for A=60⁰ and B=45⁰

=-[(tan(60⁰)+tan(45⁰))/(1-tan(60⁰)*tan(45⁰))]

=-[(sqrt(3)+1)/(1-sqrt(3)*1)] since tan(60⁰)=sqrt(3),tan(45⁰)=1

=-[(1+sqrt(3))/(1-sqrt(3))]

=-[(1+sqrt(3))/(1-sqrt(3))*((1+sqrt(3))/(1+sqrt(3)))]

=-[(1+3+2*sqrt(3))/(1-3)]

=-[(4+2*sqrt(3))/(-2)]

=-[2(2+sqrt(3))/(-2)]

=2+sqrt(3)

tan(-105⁰)=2+sqrt(3)