Q45 of 268 Page 5

If sinθ + cosθ = p and secθ + cosecθ = q, then show q(p2–1) =2p

Given: sin θ + cos θ = p and sec θ + cosec θ = q

To show q(p2 – 1) = 2p


Taking LHS = q(p2 – 1)


Putting the value of sin θ + cos θ = p and sec θ + cosec θ = q, we get


=(sec θ + cosec θ){( sin θ + cos θ )2 – 1)


=(sec θ + cosec θ){(sin2 θ + cos2 θ + 2sin θ cosθ) – 1)}


[∵ (a + b)2 = (a2 + b2 + 2ab)]


=(sec θ + cosec θ)(1+2sin θ cos θ – 1)


=(sec θ + cosec θ)(2sin θcosθ)




= 2(sin θ +cos θ)


= 2p [ given sin θ + cos θ = p]


=RHS


Hence Proved


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