Finding solution of trigonometric equation sum_(m=1)^6 cosec(theta + ((m-1) pi)/(4)) cosec(theta + (m pi)/(4)) =4 sqrt(2) if 0<theta<pi/2.

Marlene Brooks 2022-10-25 Answered
Finding solution of trigonometric equation
How to find the solutions of this trigonometric equation
m = 1 6 c o s e c ( θ + ( m 1 ) π 4 ) c o s e c ( θ + m π 4 ) = 4 2
if 0 < θ < π / 2
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Answers (1)

Kaylee Evans
Answered 2022-10-26 Author has 20 answers
HINT:
csc ( θ + ( m 1 ) π 4 ) csc ( θ + m π 4 ) = 1 sin ( θ + ( m 1 ) π 4 ) sin ( θ + m π 4 )
sin ( A B ) sin A sin B = sin A cos B cos A sin B sin A sin B = cot B cot A
Here A = θ + m π 4 and B = θ + ( m 1 ) π 4 , A B = π 4
1 sin ( θ + ( m 1 ) π 4 ) sin ( θ + m π 4 )
= 1 sin π 4 sin { θ + m π 4 ( θ + ( m 1 ) π 4 ) } sin ( θ + ( m 1 ) π 4 ) sin ( θ + m π 4 )
= cot ( θ + ( m 1 ) π 4 ) cot ( θ + m π 4 ) sin π 4
Set m = 1 , 2 , 3 , 4 , 5 , 6 and add and finally simplify.
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