Concept understanding — Composite and Sum/Difference Identities of Inverse Trigonometric Functions
These are the working-formula properties (Properties VI–X) for combining or composing inverse trig functions, together with the reference-triangle technique for a raw composite like tan(sin−1x).
Composing a trig function with an unrelated inverse trig function (reference triangle). To evaluate f(g−1(x)) where f=g (e.g. cot(sin−1x)), let θ=g−1(x), build a right triangle encoding θ from the definition of g−1 (e.g. sinθ=x gives opposite =x, hypotenuse =1, so adjacent =1−x2 by Pythagoras — taking the adjacent side non-negative since θ∈[−2π,2π] keeps cosine ≥0 there), then read f(θ) straight off the triangle. This proves, e.g., tan(sin−1x)=1−x2x, −1<x<1.
Property VI (addition/subtraction formulas).
sin−1x+sin−1y=sin−1(x1−y2+y1−x2),if x2+y2≤1 or xy<0
with analogous subtraction forms for sine and cosine. Extending the tangent addition formula to three terms gives tan−1x+tan−1y+tan−1z=tan−1[1−xy−yz−zxx+y+z−xyz]; setting the left side equal to π and taking the tangent of both sides (which is 0) proves the classical identity x+y+z=xyz whenever tan−1x+tan−1y+tan−1z=π.
Property VII (double-angle-style formulas, from setting y=x in Property VI).
Property VIII.sin−1(2x1−x2)=2sin−1x for ∣x∣≤21, and sin−1(2x1−x2)=2cos−1x for 21≤x≤1 (the SAME left side splits into two different right sides depending on which half of [−1,1], i.e. which principal-range piece, x falls in). …
Using tan−1x+tan−1y=tan−1(1−xyx+y) for xy<1 with x=41,y=92: 1−xyx+y=17/1817/36=21, so the sum equals tan−1(21). Setting θ=cos−1(53) (so cosθ=53,sinθ=54), the half-angle tangent is $\tan\dfrac{\theta}{2}=\dfrac{\sin\theta}{1+\cos\theta}=\dfrac{4/5 …