Identities similar to arctan⁡(x)+arctan⁡(1x)=π2 The arctan⁡(x)+arctan⁡(1x)=π2 identity can be solved by taking the derivative of the left hand side, showing it is 0, and then plugging in, say, x=1 to get its constant value π2 Are there any other (nontrivial) identities which can be solved similarly?

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Identities similar to arctan⁡(x)+arctan⁡(1x)=π2  The arctan⁡(x)+arctan⁡(1x)=π2 identity can be solved by taking the derivative of the left hand side, showing it is 0, and then plugging in, say, x=1 to get its constant value π2  Are there any other (nontrivial) identities which can be solved similarly?

portafilses · Accepted Answer

Answer provided below  arccos⁡(x)+arcsin⁡(x)=π2

Joy Compton · Accepted Answer

If we know sin⁡′(x)=cos⁡(x) and cos⁡′(x)=−sin⁡(x), then we can computeddx[sin2(x)+cos2(x)]=0,to conclude sin2(x)+cos2(x) is constant. Then plug in one value to getsin2(x)+cos2(x)=1

Identities similar to \arctan(x)+\arctan(1/x)=\frac{\pi}{2} The \arctan(x)+\arctan(1/x)=\pi/2 identity can be solved by

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