Correct Answer - 3
Given, ` tan theta = cot 5 theta `
` therefore " " tan theta = tan (( pi )/(2) - 5theta )`
` rArr" " (pi)/(2) - 5 theta = n pi + theta `
` rArr " " 6 theta = (pi)/(2) - npi `
` rArr" " theta = (pi)/(12)- (n pi )/(6)`
Also, ` cos 4 theta = sin 2 theta = cos ((pi)/(2) - 2 theta )`
` rArr" " 4 theta = 2 n pi pm ((pi)/(2) - 2 theta )`
Taking positive sign,
` " " 6 theta = 2 n pi + (pi)/(2)`
`rArr " " theta = ( n pi )/(3) + (pi)/(12)`
Taking negative sign ,
` 2 theta = 2 npi - (pi)/(2) rArr theta = n pi - (pi)/(4)`
Above values of ` theta ` suggest that there are only 3 common solutions.
