Prove that following identities:
P(n,n) = nP(n-1,n-1)

Prove the following identity : 2u_6-3u_4+1=0 , where u_n =cos^n theta+ sin^n theta .

Prove the following by using induction for all n in N . 1+2+3+.....+n=(n(n+1))/(2)

By the Principle of Mathematical Induction, prove the following for all n in N : a^(2n-1)-1 is divisible by a-1.

By the Principle of Mathematical Induction, prove the following for all n in N : 4^(n)-3n-1 is divisible by 9.

By the Principle of Mathematical Induction, prove the following for all n in N : 4^(n)+ 15n-1 is divisible by 9.

Prove the following: sin (n + 1)x sin (n + 2)x + cos (n + 1)x cos (n + 2)x = cos x

By the Principle of Mathematical Induction, prove the following for all n in N : 10^(2n-1)+1 is divisible by 11.

By the Principle of Mathematical Induction, prove the following for all n in N : 15^(2n-1)+1 is multiple of 16.

If x^p occurs in the expansion of (x^2+1/x)^(2n) , prove that its coefficient is : ((2n)!)/([1/3(4n-p)!][1/3(2n+p)!]) .

Prove the following by using the principle of mathematical induction for all n in N :- 1.2.3 + 2.3.4 +...+n(n+1)(n+2)=(n(n+1)(n+2)(n+3))/4 .