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).
Property IX (mixed cofunction-and-Pythagorean substitutions). For 0≤x≤1: sin−1x=cos−11−x2 and cos−1x=sin−11−x2; for −1≤x<0 both pick up a sign/−π correction; and for x>0, tan−1x=sin−11+x2x=cos−11+x21.
both proved the same way: set x=sinθ (resp. cosθ) and substitute into the ordinary triple-angle trig identity.
Tip
The domain conditions attached to Properties VI–X are not decoration — they are exactly what keeps the sum of two principal-range angles from spilling OUTSIDE the target function's own principal range (which would otherwise force an extra ±π correction). Always check the condition before trusting the formula's plain output; e.g. proving tan−1x+tan−1y+tan−1z=π⇒x+y+z=xyz works by equating tangents (valid unconditionally, since tanπ=0 regardless of branch), which is why that particular proof needs no domain restriction even though the addition formula it is built from does.
In each case, name the inner inverse-trig angle θ, build a reference right triangle from its defining ratio, then read off the requested trig ratio of θ.
✓Final answer
2x−x2.
9x2−6x+21.
1−(x+21)2x+21.
Each expression names an angle via one inverse-trig ratio, builds the other two sides by Pythagoras (always taking the positive root, matching the principal range's sign), and reads off the requested ratio.
Step 1. (i) Name the angle. Let θ=cos−1(1−x), so cosθ=1−x, θ∈[0,π], hence sinθ≥0.