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Question: In a triangle ABC, if <img src="https://cdn.pureessence.tech/canvas_494.png?top_left_x=300&top_left_...

In a triangle ABC, if then

is equal to

A

sina

B

sinB

C

sinC

D

sinA+ sinB

Answer

sinC

Explanation

Solution

The given expression is equal to

2(tan⁡( A2)+tan⁡(B2))(1−tan⁡(A2)tan⁡(B2))(1+tan⁡2( A2))(1+tan⁡2( B2))\frac { 2 \left( \tan \left( \frac { \mathrm {~A} } { 2 } \right) + \tan \left( \frac { \mathrm { B } } { 2 } \right) \right) \left( 1 - \tan \left( \frac { \mathrm { A } } { 2 } \right) \tan \left( \frac { \mathrm { B } } { 2 } \right) \right) } { \left( 1 + \tan ^ { 2 } \left( \frac { \mathrm {~A} } { 2 } \right) \right) \left( 1 + \tan ^ { 2 } \left( \frac { \mathrm {~B} } { 2 } \right) \right) }

=2sin⁡( A2+B2)cos⁡(A2+B2)=2cos⁡C2sin⁡C2=sin⁡C2 \sin \left( \frac { \mathrm {~A} } { 2 } + \frac { \mathrm { B } } { 2 } \right) \cos \left( \frac { \mathrm { A } } { 2 } + \frac { \mathrm { B } } { 2 } \right) = 2 \cos \frac { \mathrm { C } } { 2 } \sin \frac { \mathrm { C } } { 2 } = \sin \mathrm { C }