Let p(x) = cos⁻¹x - tan⁻¹x on [-1, 1]. Both cos⁻¹x and -tan⁻¹x fall as x rises…

Step 1. The direction. $\cos^{-1}x$ falls and $\tan^{-1}x$ rises, so $-\tan^{-1}x$ falls. The sum falls on the whole interval.

Step 2. The least value. At $x = 1$: $p(1) = 0 - \dfrac{\pi}{4} = -0.7854$.

Step 3. The greatest value. At $x = -1$: $p(-1) = \pi + \dfrac{\pi}{4} = \dfrac{5\pi}{4} = 3.927$.