Question

If A,B,C,D are the four angles taken in order of a cyclic quadrilateral then
the number values of x satisfying tan’x = 1+ cosA + cosB + cosC + cosD
in [-3pi ,3pi] is
(a) 6
(b) 12
(c) 7
(d) 13​

Answer 1

Given : A,B,C,D are the four angles taken in order of a cyclic quadrilateral  tan x = 1+ cosA + cosB + cosC + cosD

To find : then  the number values of x satisfying in [-3pi ,3pi]

Solution:

A,B,C,D are the four angles taken in order of a cyclic quadrilateral

Sum of opposite angles n cyclic Quadrilateral  =  180°

=>  A + C  = B + D  =  180°

=> C =   180° - A    &  D =  180° - B

tan x = 1 + cosA + cosB + cosC + cosD

=> tan x =  1 + (cosA + CosC) + (CosB + CosD)

=>  tan x =  1 + (cosA + Cos(180° - A) ) + (CosB + Cos(180° -B))

Cos (180 - x)  = -Cosx

=> tan x =  1 + (cosA  - CosA) + (CosB - Cos B )

=> tan x =  1 + 0 + 0

=> tanx  =  1

=> x =  π/4 ± nπ

=>  x  =   -11π/4 , -7π/4 , -3π/4  ,  π/4  , 5π/4 ,  9π/4  ,  

number values of x satisfying  = 6  in  [-3pi ,3pi]

option A is correct

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