if the straight line `xcosalpha+y sin alpha=p` touches the curve `x^my^n=a^(m+n)` prove that `p^(m+n)m^mn^n=(m+n)^(m+n) a^(m+n) sin^nalpha cos^malpha`

if the straight line x cos alpha+y sin alpha=p touches the curve x^(m)y^(n)=a^(m+n) prove that p^(m+n)m^(m)n^(n)=(m+n)^(m+n)a^(m+n)sin^(n)alpha cos^(m)alpha

If the straight line x cosalpha+y sinalpha=p touches the curve x^my^n =a^(m+n) , prove that p^(m+n).m^m.n^n =(m+n)^(m+n)a^(m+n)cos^m alphasin^nalpha

If the straigjht line x cos alpha+ y sin alpha=p touches the curve x^(m)y^(n)=a^(m+n) , prove that. p^(m+n)""m^(m)n^(n)=(m+n)^(m+n)a^(m+n)sin^(n) alpha cos^(m) alpha

If the line x cos alpha+y sin alpha= p touches the curve x^my^n= a^(m+n) prove that p^(m+n).m^m.n^n= a^(m+n)(m+n)^(m+n).cos^m alpha. Sin^n alpha .

If x = cos alpha + i sin alpha , y = cos beta + i sin beta , prove that x^(m)y^(n)+1/(x^m y^n) = 2 cos (m alpha + n beta) .

If m, n integers and x = cos alpha + i sin alpha , y = cos beta + i sin beta, then prove that x^(m) y^(n) + 1/(x^(m)y^(n)) = 2 cos ( m alpha + n beta) " and " x^(m) y^(n) - 1/(x^(m) y^(n)) = 2 i sin ( m alpha + n beta) .

If m cos alpha-n sin alpha=p then prove that m sin alpha+n cos alpha=+-sqrt(m^(2)+n^(2)-p^(2))

If in an HP, t_m=n and t_n=m then prove that t_(m+n)=(mn)/(m+n) .

For as Ap. S_(n)= m and S_(m) = n . Prove that S_(m+n)=-(m+n).(m ne n)

If l,m and n are three straight lines such that l||m and l||n then prove that m||n .