You could try to determine the TIFF Compression Tag. Since black and white tiffs use a different compression algorithm. But this way you can not be 100 percent sure. If you want to be 100% sure you would need something like Exiftool. There is a library you could use from java
To be sure that the image is pure black and white you can use java.awt.image.BufferedImage and loop through the pixels. Then get the RGB value by using the method int getRGB(int x, int y). When the colors are pure black or white the getRGB should return either 0 or 255 when another value is returned the image is not pure black and white. Keep in mind that when using large images this will consume large amounts of memory.