The absolute value of the expression `tanpi/(16)+tan(5pi)/(16)+tan(9pi)/(16)+tan(13pi)/(16)` is ______

Answer

4

Explanation

Let `theta = (pi)/(6) or 8 theta = (pi)/(2)`
`y= tan theta + tan 5 theta + tan 9 theta + tan 13 theta`
`therefore y = (tan theta - cot theta)+ (tan 5 theta - cot 5 theta)` [As `tan 13 theta = tan ( 8theta + 5 theta) =- cot 5 theta and tan 9theta = tan (8theta + theta) =- cot theta `]
`= (tan theta - cot theta) + ( cot 3 theta - tan 3 theta)`
`= ( sin ^(2) theta - cos ^(2) theta)/( sin theta cos theta) + ( cos ^(2) 3 theta - sin ^(2) 3 theta)/( sin 3 theta cos 3 theta ) `
`rArr y =2 [(cos 6 theta)/( sin 6theta) - (cos 2 theta)/( sin 2theta)] `
`" " = 2[(sin 2 theta cos 6 theta - cos 2 theta sin 6 theta )/(sin 6 theta sin 2 theta)] `
`= - 2 [ ( sin 4 theta)/(cos 2 theta sin 2 theta)] = -4 " " ( because theta = (pi)/(2) - 2 theta)`
Hence, absolute value is 4.