In any triangle ABC, which is not right angled ∑ cos A ⋅ textcosecB ⋅ textcosec C is equal to

Question

In any triangle ABC, which is not right angled ∑ cos A ⋅ textcosecB ⋅ textcosec C is equal to (A) 1 (B) 2 (C) 3 (D) 0

Tardigrade Admin · Accepted Answer

∑ cos A textcosec B textcosec C=( cos A/ sin B sin C)+( cos B/ sin A sin C)+( cos C/ sin A sin B) =( cos A sin A+ cos B sin B+ cos C sin C/ sin A sin B sin C) =( sin 2 A+ sin 2 B+ sin 2 C/2 sin A sin B sin C)=(4 sin A sin B sin C/2 sin A sin B sin C)=2 (using conditional identity)

Q. In any triangle ABC, which is not right angled ∑cosA⋅cosecB⋅cosecC is equal to

1

2

3

0

∑cosAcosecBcosecC=sinBsinCcosA​+sinAsinCcosB​+sinAsinBcosC​ =sinAsinBsinCcosAsinA+cosBsinB+cosCsinC​ =2sinAsinBsinCsin2A+sin2B+sin2C​=2sinAsinBsinC4sinAsinBsinC​=2 (using conditional identity)

Q. In any triangle $A BC$ , which is not right angled $\sum cos A \cdot cosecB \cdot cosec C$ is equal to