Class 12 Inverse Trigonometric Functions · Long Answer3 Marksinverse_trigonometric_identities
Class 12 Maths · Inverse Trigonometric Functions · Long Answer #105
If \(\tan^{-1}a+\tan^{-1}b+\tan^{-1}c=\pi\), prove a+b+c=abc.
✅ Step-by-Step Solution
Final Answer: a+b+c=abc
- Let A=arctan a, B=arctan b, C=arctan c, so A+B+C=π and each cosine is positive.
- Expand sin(A+B+C)=sin A cos B cos C+cos A sin B cos C+cos A cos B sin C−sin A sin B sin C.
- The sum angle is π, so the expansion is 0. Divide by the nonzero product cos A cos B cos C to get 0=tan A+tan B+tan C−tan A tan B tan C.
- Substituting a,b,c yields a+b+c=abc.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Let A=arctan a, B=arctan b, C=arctan c, so A+B+C=π and each cosine is positive.
- [1 mark] Expand sin(A+B+C)=sin A cos B cos C+cos A sin B cos C+cos A cos B sin C−sin A sin B sin C.
- [1 mark] The sum angle is π, so the expansion is 0. Divide by the nonzero product cos A cos B cos C to get 0=tan A+tan B+tan C−tan A tan B tan C.
- [0.5 marks] Substituting a,b,c yields a+b+c=abc.
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