Cochineal (carmine, carminic acid, E120, 120, or C.I. 75470) doesn't seem to be the same thing as:
Allura red (E129, 129, C.I. 16035, food red 17, FD&C red 40), which is an artificial dye introduced in the US in 1971 to replace:
Amaranth (E123, 123, C.I. 16185, FD&C red 2), which is a synthetic coal tar dye and azo dye (and obviously nothing to do with the plant, amaranth). It was banned in 1976 in the US, because it was found to produce malignant tumours in female rats.
There was a short item on an Australian TV news show earlier this month, which linked 129 (Allura red, or your Red #40) with an increased risk of cancer. The story said that the dye produced tumours in mice, and that it has been banned in several countries of Europe, including the UK, Germany and France.
But, interestingly enough, the book, The New additive code breaker, says:
"Allura red has been shown to be non-genotoxic. It does not cause cancer in rats or mice, but para-cresidine, a compound used in the production of the dye, causes bladder tumours in rodents. No free para-cresidine has been found in food-grade Allura red."
Of course that was written in 1989, so there may have been further tests since then which did find that the dye can cause cancer.
I don't like the sound of any of those three red dyes, to be honest, so I tend to avoid any products that are *dyed* red.