Are $\cos^2 \theta$ and $\cos \theta^2$ the same?
Are $\cos^2 \theta$ and $\cos \theta^2$ the same? I mean be it $\sin,\cos, \tan ,\cot ,\sec,\csc$. Are they same? Please help a maths noob here.
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$\begingroup$No, they are not the same.
When you type $\cos^2 \theta$ you actually mean $(\cos \theta)^2$.
When you type $\cos \theta^2$ you mean $\cos(\theta ^2)$.
$\endgroup$ 3 $\begingroup$The first notation is used to mean $$\cos^2 \theta = \left( \cos \theta \right)^2$$ Your second notation will usually be read as $$\cos \theta^2 = \cos \left( \theta^2 \right)$$ although it is sometimes preferred to use the notation in the right-hand side to be clear.
They are not the same since $$\left( \cos \theta \right)^2 = \cos\theta\cos\theta \ne \cos(\theta\theta) = \cos(\theta^2)$$
$\endgroup$ 6 $\begingroup$It's a matter of syntax.
In handwriting or literature: $\sin^{2} x \equiv (\sin x)^{2}$ and $\cos x^{2} \equiv \cos (x^{2})$;
however for computer software, say Mathematica:
Cos[x]$^2$ refers to $(\cos x)^2$ while Cos[x$^2$] refers to $\cos (x^{2})$.
Should be very careful.
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