Prove that following identities:
`C(n,2) = 1/2 n (n-1)`

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

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

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 : 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 (1+x)^n =C_0+C_1 x+ C_2 x^2 +....... C_nx^n prove the following : C_0C_n+C_1C_(n-1)+C_2C_(n-2)+.....+ C_nC_0= ((2n!))/(n!)^2 .

If (1+x)^n =C_0+C_1 x+ C_2 x^2 +....... C_nx^n prove the following : 2C_0+5 C_1+8C_2+...+(3n+2)C_n=(3n+4)2^(n-1) .

By the Principle of Mathematical Induction, prove the following for all n in N : 3^(4n+1)+2^(2n+2) is divisible by 7.

If (1+x)^n =C_0+C_1 x+ C_2 x^2 +....... C_nx^n prove the following : C_0-C_1/2+C_2/3-.....+(-1)^n C_n/(n+1)= 1/(n+1) .