Question

In triangle ABC, angle C=90, then tan A+ tan B=
​

Answer 1

Answer:

The triangle is an obtuse angled triangle.

We know,

tan(A+B+C)=tanA+tanB+tanC−tanAtanBtanC1−tanAtanB−tanAtanC−tanBtanCtan⁡(A+B+C)=tan⁡A+tan⁡B+tan⁡C−tan⁡Atan⁡Btan⁡C1−tan⁡Atan⁡B−tan⁡Atan⁡C−tan⁡Btan⁡C

A+B+C=πA+B+C=π

⟹tan(A+B+C)=0⟹tan⁡(A+B+C)=0

⟹tanA+tanB+tanC=tanAtanBtanC⟹tan⁡A+tan⁡B+tan⁡C=tan⁡Atan⁡Btan⁡C

⟹tanAtanBtanC<0⟹tan⁡Atan⁡Btan⁡C<0

Now we know that the tantan function is negative in the second quadrant, so for the above condition to be satisfied there has to be one angle >90o>90o. Hence we can conclude that the triangle is an obtuse one.