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Examples for

Step-by-Step Proofs

Trigonometric Identities

See the steps toward proving a trigonometric identity:

does sin(θ)^2 + cos(θ)^2 = 1?
(1 + tan(x))/(1 - tan(x)) = (cos(x) + sin(x))/(cos(x) - sin(x))
cot(t/2)^2 = (1 + cos(t)) / (1 - cos(t))
verify tanθ + cotθ = secθ cscθ

Mathematical Induction

Prove a sum or product identity using induction:

prove by induction sum of j from 1 to n = n(n+1)/2 for n>0
prove sum(2^i, {i, 0, n}) = 2^(n+1) - 1 for n > 0 with induction
prove by induction product of 1 - 1/k^2 from 2 to n = (n + 1)/(2 n) for n>1

Prove divisibility by induction:

using induction, prove 9^n-1 is divisible by 4 assuming n>0
induction 3 divides n^3 - 7 n + 3

Prove an inequality through induction:

show with induction 2n + 7 < (n + 7)^2 where n >= 1
prove by induction (3n)! > 3^n (n!)^3 for n>0

Prove a sum identity involving the binomial coefficient using induction:

prove by induction sum C(n,k) x^k y^(n-k),k=0..n=(x+y)^n for n>=1
prove by induction sum C(n,k), k=0..n = 2^n for n>=1

RELATED EXAMPLES

  • Trigonometry
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