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Parity and the Pythagorean identity for sine and cosine
Statement
For every real , Consequently and .
Facts & Assumptions
Given: A real .
For all real , and (The addition formulas for sine and cosine).
The derivative identities and values at zero hold (The derivatives of sine and cosine are cosine and minus sine).
The algebra of derivatives and the zero-derivative theorem hold (Sums, scalar multiples, products and quotients: , , , and when , A function continuous on an interval whose derivative vanishes at every interior point of is constant on ; consequently two such functions with the same derivative differ by a constant).
Differentiability implies continuity (A function differentiable at is continuous at ).
Proof
The derivative of is .
Step 1.1 makes differentiable everywhere, hence continuous by [L4]. The zero-derivative theorem therefore makes constant, and , proving ; each square is then at most .
Applying the addition formulas at gives The coefficient matrix squares to the identity by step 2.1, so these equations give and .
Depends on
- The addition formulas for sine and cosine
- The derivatives of sine and cosine are cosine and minus sine
- Sums, scalar multiples, products and quotients: $(f+g)'(c) = f'(c) + g'(c)$, $(\alpha f)'(c) = \alpha f'(c)$, $(fg)'(c) = f'(c)g(c) + f(c)g'(c)$, and $(f/g)'(c) = \bigl(f'(c)g(c) - f(c)g'(c)\bigr)/g(c)^{2}$ when $g(c) \ne 0$
- A function differentiable at $c$ is continuous at $c$
- A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$; consequently two such functions with the same derivative differ by a constant
Used by
- Polar coordinates on a full closed angular period are not injective and are singular at radius zero Counterexample
- Principal inverse sine and inverse cosine Definition
- Cylindrical coordinates have absolute Jacobian determinant r on an injective compact box Example
- Polar change of variables on a compact annular sector gives the Jacobian factor r and its area Example
- Spherical coordinates have absolute Jacobian determinant r² sinφ away from the axis and angular seam Example
- The extension of x² sin(1/x) by zero is differentiable but its derivative is discontinuous at zero Example
- The hyperspherical-coordinate Jacobian is the standard product of a radial power and sine powers Example
- x sin(1/x) tends to zero despite its oscillation Example
- Derivatives and fundamental periods of tangent, cotangent, secant, and cosecant Theorem
- Double-angle and quadratic power-reduction identities Theorem
- For -1<y<1, (arcsin y)ᵖʳⁱᵐᵉ=1/√1-y² and (arccos y)ᵖʳⁱᵐᵉ=-1/√1-y² Theorem
- Pythagorean and parity identities for all six trigonometric functions on their natural domains Theorem
- Quarter-turn values and shifts by pi/2 and pi Theorem
- t↦(cos t,sin t) is a bijection from [0,2π) onto the real unit circle Theorem
- The subtraction formulas for sine and cosine Theorem
- The tangent half-angle identities and rational parametrization of the unit circle away from (-1,0) Theorem
- Tₙ(cosθ)=cos(nθ) and Uₙ(cosθ)sinθ=sin((n+1)θ) for every n∈ℕ Theorem
- Triple-angle identities for sine, cosine, and tangent Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 59 results over 22 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- NIST Digital Library of Mathematical Functions, Chapter 4 (standard reference, not scraped)
- C. Schmeiser, Introduction to Analysis (standard reference, not scraped)