Alan

$$\\
\frac{\cos(x)\sin^2(x)}{1-\cos(x)}=\cos(x)+\cos^2(x) \\\\
\text{Divide by cos(x)}\\
\frac{\sin^2(x)}{1-\cos(x)}=1+\cos(x) \\\\
\text{Multiply by 1-cos(x)}\\
\sin^2(x)=(1-\cos(x))(1+\cos(x))\\\\
\text{The right-hand side is the difference of two squares so}\\
\sin^2(x)=1-\cos^2(x)\\\\
\text{so:}\\
\sin^2(x)+\cos^2(x)=1\\
\text{which is correct!}$$

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Use the sine rule:  sin(A)/a = sin(B)/b etc.

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The inequality holds when -4 < x < 2

Look at the graph below and think carefully about what happens in each region:

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You are correct Melody and xvxvxv.

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No.  The cross-sectional area is the area perpendicular to the direction of current flow, through which the current is flowing.  If the current were flowing axially through a solid cylinder of radius r, then the cross-sectional area would, indeed, be pi*r², and it would be constant all the way along the cylinder.  However, here the current is flowing radially outward , so the cross-sectional area gets bigger as you go further out from the centre.

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I read this as an attempt to do a straight-edge and compass construction (as per the ancient Greeks).  However, the angle indicated isn't in fact 20° - see image below:

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6x10²³ atoms of silicon weigh 28 grams, so one atom of silicon weighs 28x10-²³/6 grams ≈ 4.7x10^-23 grams

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Start with the large equilateral triangles - they are easy to construct with straight-edge and compass.  Then you can draw the larger circle easily enough - you don't need to use a 20° angle.  The four smallest equilateral triangles are also easy to construct (I'm assuming the four intermediate equilateral triangles created by the intersection of the two large triangles are themselves equal in size to four of the smallest triangles.

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Therre are approximately 6x10²³ molecules in 44 grams of carbon dioxide (44 is molecular weight of CO₂), so just multiply this by 34/44

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