Which approach is better for my APS-C Canon 750D dSLR?
Capturing images in JPEG and an aspect ratio as 1:1 or 3:2? (for portrait and landscape photography)
Capturing images in RAW format and later cropping in Lightroom?
As the sensor is 3:2 that's the natural size to choose when shooting. Cropping can be done in post, where you have any option you want. If you crop when you shoot you waste pixels. In my experience it is best to crop later if possible and to capture as much as possible - sometimes you find a composition ( framing ) in post you were not expecting when you shoot. I also advocate framing a little loosely when you shoot, as this makes post production easier and gives you more scope for correcting errors.
You have the option on the 750D to shoot RAW+JPEG ( simultaneously ), so it is not an either or.
In general JPEG alone on DSLRs sometimes offers faster or longer continuous shooting, while RAW allows you the maximum latitude to adjust images later.
An image should be captured at the native aspect-ratio of the sensor which is 3:2. You will capture everything you saw in the viewfinder, plus a 5% more in your case since the OVF on the 750D only provides 95% coverage.
You will later be able to crop to your liking, including other aspect-ratios and other positions. Should you crop in-camera, you will be forced on a certain aspect-ratio and symmetric crop around the center which is not always desirable.
RAW images are always uncropped, so use that format if that is what you like but don't stop using JPEGs if that is more convenient to you just because of the cropping difference.
If you know you are aiming for a result in a certain aspect ratio (say, square, or 8x10), and your camera supports showing you that format as you shoot, that can be a huge advantage. With a DSLR, where the viewfinder is generally a physical thing made to match the same (or similar) view as will be captured by the sensor, this is less useful — but the option may be available in live view.
If your camera allows JPEG plus RAW capture, this gives you the best of both worlds: you can compose for your intended end result, and get out-of-camera results which may be just what you need, or, if you change your mind, the RAW file will contain the full, uncropped frame.
In my opinion I would always use the full resolution (3:2). A crop is a simple enough process to do after. (I had never seing a DSLR camera having an aspect ratio of 1:1)
If you crop your image when capturing you do not see that crop on your viewfinder, which is eventually the reason why you choose a DSLR in the first place.
Regarding the file format (raw vs jpeg) your choice is determined by:
1) You do want to adjust it later. ("later" could be years after).
2) You want to use them dicrectly from the camera.
Aspect ratio is the width to height proportions of the image. This is also called the “format”. The modern digital systems are structured after the venerable 35mm film frame which measures 24mm height by 36mm length. The aspect ratio is 36 ÷ 24 = 1.5 (also called 3:2 as 3÷2 =1.5). Now for the rest of the story:
The American inventor Thomas Edison came up with an idea to make a motion picture viewer as a money maker to be installed in penny arcades. This he called the Kinetoscope, patented in 1891. Edison and his engineers W. K. L. Dickers and William Heise bargained Kodak for film and purchased rolls of 70mm wide film. Kodak was making this roll film for the Brownie camera. Edison had this team slit the roll in half making two 35mm wide rolls for the price of one.
The Kinetoscope system required sprocket holes along the film edges. These engaged the camera’s film transport mechanism. With the sprocket holes punched, the width remaining for the image measured 24mm for the image width and the height was set to 18mm. For many years this remained the standard format of motion pictures. As time passed, motion pictures flourished as did 35mm film manufacturing.
Ernest Leitz and his Chief engineer Oskar Barnack invented a still camera, the Leica, marketed in 1924. This camera utilized surplus 35mm motion picture film that was plentiful. The camera was designed to be held in the landscape positon; thus the film feed was horizontal. He retained the 24mm and doubled the 18mm to 36mm.Thus the format was 24mm height by 36mm length. This is the 1.5 aspect ratio we know and love.
The aspect ratio of 1.5 has an earlier origin in art. Mystics and mathematicians conclude that a ratio of 3 to 5 was artistically pleasing. This was used by the Egyptians and the artists of the renaissance. This is a ratio of 1.6. Today’s digitals are commonly 1.5 or 1.6 as to aspect ratio.
I predict that today’s wide-screen HD TV’s with an aspect ratio of 16:9 will prevail and that future digital still cameras will adopt this ratio.