$Simplify this expression$

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I've attached a file with the expression screenshooted. The answer is supposed to be (1 + 2 sin x cos x)? ?1. But I can't arrive at it. Help would be appreciated.

Trigonometry

Lucija Karalic

Matchmaticians Simplify this expression File #1

File #1 (png)

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Savionf

Does " : " means division here?

Answer 1

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We have
\[\frac{\sin^{-2}x+\cos^{-2}x}{(\sin^{-1}x+\cos^{-1}x)^2}=\frac{\frac{1}{\sin^{2}x}+\frac{1}{\cos^{2}x}}{(\frac{1}{\sin x}+\frac{1}{\cos x})^2}\]
\[=\frac{\frac{\cos^2 x +\sin^2 x}{\sin^{2}x \cos^2 x}}{(\frac{\cos x+\sin x}{\sin x \cos x})^2}=\frac{\frac{1}{\sin^{2}x \cos^2 x}}{\frac{(\cos x+\sin x)^2}{\sin^2 x \cos^2 x}}\]
\[=\frac{1}{(\sin x+\cos x)^2}=\frac{1}{\sin^2 x+2\sin x \cos x+\cos^2 x}\]
\[=\frac{1}{1+2\sin x \cos x}=(1+2\sin x \cos x)^{-1}.\]

Savionf

$Simplify this expression$

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