Prove that
`sin^(2)phi(1+cot^(2)phi)=1`

If phi_(1) and phi_(2) be the apparent angles of dip observed in two vertical planes at right angles to each other , then show that the true angle of dip phi is given by cot^(2) phi = cot^(2) phi_(1) + cot^(2) phi_(2) .

Prove that ( sin 2 A )/( 1 - cos 2 A) = cot A

If sin theta is G.M. of sin phi and cos phi, then prove that cos 2 theta = 2 cos ^(2) ((pi)/(4) + phi).

If theta+phi+psi=2pi, prove that cos^2theta+cos^2phi+cos^2psi -2costheta cosphi cospsi=1

If phi (x) is a differentiable real valued function satisfying phi (x) +2phi le 1 , then it can be adjucted as e^(2x)phi(x)+2e^(2x)phi(x)lee^(2x) or (d)/(dx)(e^(2)phi(x)-(e^(2x))/(2))le or (d)/(dx)e^(2x)(phi(x)-(1)/(2))le0 Here e^(2x) is called integrating factor which helps in creating single differential coefficeint as shown above. Answer the following question: If p(1)=0 and dP(x)/(dx)ltP(x) for all xge1 then

Prove: (sin phi - cos phi) ^(2) =1 - sin 2 phi.

Prove that cos 2 theta cos 2 phi + sin ^(2) ( theta - phi) -sin ^(2) (theta + phi) = cos ( 2 theta + 2 phi).

Prove that for a null set phi, n{P(P(phi))}=2 .

If cos(theta+phi)= mcos(theta-phi) , then prove that tan theta=(1-m)/(1+m)cotphi .

Prove: (1+cot^2A)sin^2A=1