Question

tan theta + 3 cot theta=5 sec theta

Answer 1

The equation can be rewritten as 

(sin(theta)/cos(theta)) + 3*(cos(theta)/sin(theta)) = 5*(1/cos(theta)) ----> 

(sin^2(theta) + 3*cos^2(theta)) / (sin(theta)*cos(theta)) = 5*(1/cos(theta)) ------> 

sin^2(theta) + 3*(1 - sin^2(theta)) = 5*sin(theta) ----> 

2*sin^2(theta) + 5*sin(theta) - 3 = 0 ----> 

(2*sin(theta) - 1)(sin(theta) + 3) = 0, so either sin(theta) = 1/2 or sin(theta) = -3. 

But since the range of the sine function is [-1,1] the only solution is 

sin(theta) = 1/2, which has solutions on [0,2pi] of theta = pi/6 and theta = 5pi/6. 

Note that in the third step I canceled cos(theta) from both sides, which I could

do only by ruling out cos(theta) = 0 as a potential solution. (In the end it is not 

a solution anyway.)