Theorem tanhaddd

Description: Addition formula for tanh. (Contributed by Jon Pennant, 9-Sep-2026)

	Ref	Expression
Hypotheses	tanhaddd.a	$$⊢\phi \to A\in ℂ$$
	tanhaddd.b	$$⊢\phi \to B\in ℂ$$
	tanhaddd.1	$$⊢\phi \to \cosh \left(A\right)\ne 0$$
	tanhaddd.2	$$⊢\phi \to \cosh \left(B\right)\ne 0$$
	tanhaddd.3	$$⊢\phi \to \cosh \left(A+B\right)\ne 0$$
Assertion	tanhaddd	$$⊢\phi \to \tanh \left(A+B\right)=\frac{\tanh \left(A\right)+\tanh \left(B\right)}{1+\tanh \left(A\right)\tanh \left(B\right)}$$

Proof

Step	Hyp	Ref	Expression
1		tanhaddd.a	$$⊢\phi \to A\in ℂ$$
2		tanhaddd.b	$$⊢\phi \to B\in ℂ$$
3		tanhaddd.1	$$⊢\phi \to \cosh \left(A\right)\ne 0$$
4		tanhaddd.2	$$⊢\phi \to \cosh \left(B\right)\ne 0$$
5		tanhaddd.3	$$⊢\phi \to \cosh \left(A+B\right)\ne 0$$
6	1 2	jca	$$⊢\phi \to \left(A\in ℂ\wedge B\in ℂ\right)$$
7		tanhadd	$$⊢\left(\left(\left(A\in ℂ\wedge B\in ℂ\right)\wedge \cosh \left(A\right)\ne 0\right)\wedge \left(\cosh \left(B\right)\ne 0\wedge \cosh \left(A+B\right)\ne 0\right)\right)\to \tanh \left(A+B\right)=\frac{\tanh \left(A\right)+\tanh \left(B\right)}{1+\tanh \left(A\right)\tanh \left(B\right)}$$
8	6 3 4 5 7	syl22anc	$$⊢\phi \to \tanh \left(A+B\right)=\frac{\tanh \left(A\right)+\tanh \left(B\right)}{1+\tanh \left(A\right)\tanh \left(B\right)}$$