17.3 Osborn’s Rule

The rule is to replace each trigonometric function by the corresponding hyperbolic function, and to change the sign of every product — or implied product — of two sines.

Note 17.5. “Implied product” is the part that catches people. \(\tan ^{2}x\) has \(\sin ^{2}x\) hidden inside it, since \(\tan ^{2}x=\frac {\sin ^{2}x}{\cos ^{2}x}\), so it counts as a product of two sines and its sign must change. The same goes for \(\cot ^{2}x\) and \(\operatorname {cosec}^{2}x\). Terms such as \(\cos ^{2}x\) or \(\sec ^{2}x\) contain no sines and are left alone.

Applied to \(\cos ^{2}x+\sin ^{2}x=1\): the \(\sin ^{2}x\) is a product of two sines, so its sign flips and we get \(\cosh ^{2}x-\sinh ^{2}x=1\), the first identity above. Applied to \(\sec ^{2}x=1+\tan ^{2}x\), the hidden \(\sin ^{2}x\) flips, giving \(\operatorname {sech}^{2}x=1-\tanh ^{2}x\), which is the sixth identity rearranged.

The rule is a memory aid, not a proof. It works because the hyperbolic functions are the trigonometric ones with \(x\) replaced by \(ix\), and each sine so converted contributes a factor of \(i\), so two of them contribute \(i^{2}=-1\). Anything derived from it should be checked against the definitions if it matters.

Example 17.6.

Write the equivalent hyperbolic identity of the trigonometric identity

1.: \(\cos 2x=1-2\sin ^2x\)
2.: \(\tan (A-B)=\frac {\tan A-\tan B}{1+\tan A\tan B}\)

Sample Questions

1.

Prove the following identities:

(a): \(\cosh A=2\cosh ^2A+1\)
(b): \(\tanh ^2A+\text {sech}^2A=1\)
(c): \(\cosh (A-B)=\cosh A \cosh B-\sinh A \sinh B\)
(d): \(\cosh A+\cosh B =2\cosh \frac {(A+B)}{2}\cosh \frac {(A-B)}{2}\).

2.

Given that \(x>0\), show that \(\sinh (\ln x)=\frac {x^2-1}{2x}\)

3.

Given that \(y=\ln \left [x+(1+x^2)^{\displaystyle {\frac {1}{2}}}\right ]\) by differentiation show that \([1+x^2]\left (\frac {dy}{dx}\right )^2=1\).

4.

Solve \(2\cosh x +\sinh x=2\)

5.

Find the value of \(x\) given that \(\tanh x=\frac {1}{2}\).

6.

Solve for real values of \(x\) if \(\cosh x\sinh x=\frac {3}{2}\).

7.

Find the derivatives of the function \(f(x)=e^{\displaystyle {\tanh ^2x}}\).

8.

Find the real values of \(x\) such that \(\cosh ^2x-1=-3\).

9.

Differentiate \(y=x\tanh \sqrt {x}\).

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