How can you prove that #1 - tanh^2x = sech^2x#?
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To prove that (1 - \tanh^2x = \operatorname{sech}^2x), we start with the definition of hyperbolic tangent and hyperbolic secant:
[\tanh x = \frac{{\sinh x}}{{\cosh x}}]
[\operatorname{sech} x = \frac{1}{{\cosh x}}]
Now, we square the expression for hyperbolic tangent:
[\tanh^2 x = \left(\frac{{\sinh x}}{{\cosh x}}\right)^2 = \frac{{\sinh^2 x}}{{\cosh^2 x}}]
Using the identity (\cosh^2 x - \sinh^2 x = 1), we can rewrite (\sinh^2 x) as (\cosh^2 x - 1):
[\tanh^2 x = \frac{{\cosh^2 x - 1}}{{\cosh^2 x}} = 1 - \frac{1}{{\cosh^2 x}} = 1 - \operatorname{sech}^2 x]
Therefore, (1 - \tanh^2 x = \operatorname{sech}^2 x).
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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