Sin theeta plus cos theeta is equal to p and sec theeta plus cosec theeta is equal to q then prove q int p sq. Minus1 is equal to 2p
Answers
Given : sin θ + cos θ = p
sec θ + cosec θ = q
To Prove : q × (p² - 1) = 2p
Proof : We solve RHS and LHS separately.
RHS = 2p
= 2 ( sin θ + cos θ )
LHS = q × (p² - 1)
= (sec θ + cosec θ) × [ (sin θ + cos θ)² - 1 ]
= (sec θ + cosec θ) × [ (sin² θ + cos² θ + 2 sinθ cosθ) - 1 ]
= (sec θ + cosec θ) × ( 1 + 2 sinθ cosθ - 1 )
= (sec θ + cosec θ) × ( 2 sinθ cosθ )
= [ ( 1/ cos θ ) + ( 1/ sin θ ) ] × ( 2 sinθ cos θ )
= 2 sinθ + 2 cosθ
= 2 ( sin θ + cos θ )
LHS = RHS
Hence, proved
Answer:
L.H.S
q(p^2-1)
= (sectheeta+cosectheeta)×[(sintheeta+costheeta)^2-1]
= (sectheeta+cosectheeta)×[(sin^2theeta+2sintheeta.costheeta+cos^2theeta)-1]
= (sectheeta+cosectheeta)×(1+2sintheeta.vostheeta-1)
= (sectheeta+cosectheeta)+2sintheeta.costheeta)
=(1/costheeta+1/sintheeta)×(2sintheeta.costheeta)
=2sintheeta+2costheeta
=2(sintheeta+costheeta)
(sintheeta+costheeta=p)
=2p is answer