Question

cos20°-sin20°=P then cos40°= ?

Answer 1

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⇒ cos 20 - sin 20=p
square it
cos220 + sin220 -2sin20cos20 =p2
sin40 =1-p2
and cos40 =(1-sin240)1/2  =(2p2-p4)1/2
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Answer 2

Therefore the value of $cos40=\sqrt{2P^{2} -P^{4} }$

Step-by-step explanation:

Given;

$cos20-sin20=P$ ---equation-1

Then $cos40=?$

Squaring on both sides in equation-1;

  $(cos20-sin20)^{2} =P^{2}$

⇒$cos^{2} 20+sin^{2}20-2cos20 sin20=P^{2}$

⇒$2cos20sin20=1-P^{2}$   (∵$cos^{2} 20+sin^{2}20=1$)

⇒$sin40=1-P^{2}$  (∵$2sinAcosA=sin2A$)

⇒$sin^{2} 40=(1-P^{2} )^{2}$

We know;

$cos^{2}40=1-sin^{2}40$

⇒$cos^{2}40=1-(1-P^{2} )^{2}$

⇒$cos40=\sqrt{1-1+2P^{2} -P^{4} }$

∴$cos40=\sqrt{2P^{2} -P^{4} }$

So the value of $cos40=\sqrt{2P^{2} -P^{4} }$