How to prove that \[ \frac12\frac1{1+\sin^2 x} + \frac12\frac1{1+\cos^2 x} + \frac12\frac1{1+\sec^2 x}+ \frac12\frac1{1+\csc^2 x} = 1? \] Some genius please help me I have been stuck at this for one whole day.
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Observe that for any $\alpha> 0$ we have $$ \frac{1}{1+\alpha}+\frac{1}{1+1/\alpha}=\frac{1+1/\alpha+1+\alpha}{(1+\alpha)(1+1/\alpha)}= 1 $$ so $$ \frac{1}{1+\sin^2x}+\frac{1}{1+\csc^2x} = 1 $$ wherever $\csc x$ is defined and similarly $$ \frac{1}{1+\cos^2x}+\frac{1}{1+\sec^2x} = 1 $$ |
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Forget about the $2$'s for a while. We have $$\frac{1}{1+\sec^2 x}=\frac{1}{1+\frac{1}{\cos^2 x}}=\frac{\cos^2 x}{\cos^2 x+1}.$$ Now add $\frac{1}{1+\cos^2 x}$. We get something very simple. Continue. |
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