sin theta +cos theta =p and secsquare theta +cosec square theta =q then prove that q(p square -1) =2p

SinA + CosA = p

SecA + CosecA = q

Now, p2 - 1

= (SinA + CosA)2 - 1

= (Sin2A + 2SinACosA + Cos2A) - 1

= [ (Sin2A+Cos2A) + 2SinACosA ] - 1

= [ 1 + 2SinACosA ] - 1

= 2SinACosA [Equation (i)]

So, q(p2 - 1)

= (SecA + CosecA)*(2SinACosA) [from Equation (i)]

= SecA*(2SinACosA) + CosecA*(2SinACosA)

= (1/CosA)*(2SinACosA) + (1/SinA)*(2SinACosA)

= 2SinA + 2CosA

= 2(SinA + CosA)

= 2p

Regards

Arun (askIITians forum expert)