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