Reason that it's clicking is that you are jumping from 1 to 0 in the time that the audio signal reads the last of the 1 control signal and then the 0 control signal.
You have to slow fast signal transitions over multiple samples, for example 1 to 0 should take 5ms or 10ms which is dozens or 100ds of samples.
To do that, you can use a lowpass filter at about 300Hz more or less, which can actually be very cheap for a single monophone LFO its less that 1% cpu useage, or you can use a signal smoothing algorithm, for exampel a linear interpolation function or a SmoothStep which is awesome general knowledge for audio programmers.
BTW what is the platform? i can't give you code exampels if i dont know what code you use.
the cool thing about smoothstep is how easy it is, its very good knowldedge in synths:
check the filtered sawtooth there.
You can also a smoother that says "if difference between last signal and this signal was larger than 0.02, signal - 0.02... and create a secondary smothed signal... there are fairly simple smoothing algo's based on that it's pretty straightforward.
There is also an entire page on the topic.