Verify the hyperbolic function
1 - tanh^2x = sech^2x
if you believe cosh^2x - sinh^2x = 1 then just divide through by cosh^2x
If not, then you can verify by using the definitions of
coshx = (e^x + e^-x)/2
sinhx = (e^x - e^-x)/2
To verify the given hyperbolic function equation 1 - tanh^2x = sech^2x, we will use the definitions of the hyperbolic functions and algebraic manipulations.
1. Start with the left-hand side (LHS) of the equation:
LHS = 1 - tanh^2x
2. Use the definition of tanh(x):
tanh(x) = sinh(x) / cosh(x)
3. Substitute the definition of tanh(x) into the LHS:
LHS = 1 - (sinh(x) / cosh(x))^2
4. Simplify by squaring the expression inside the parentheses:
LHS = 1 - sinh^2(x) / cosh^2(x)
5. Use the identity relating sinh(x) and cosh(x):
cosh^2(x) - sinh^2(x) = 1
Rearrange the equation:
sinh^2(x) = cosh^2(x) - 1
6. Substitute the rearranged equation into the LHS:
LHS = 1 - (cosh^2(x) - 1) / cosh^2(x)
7. Simplify the expression inside the parentheses:
LHS = 1 - (cosh^2(x) - cosh^2(x) + 1) / cosh^2(x)
8. Combine similar terms:
LHS = 1 - 1 / cosh^2(x)
9. Use the definition of sech(x):
sech(x) = 1 / cosh(x)
10. Substitute the definition of sech(x) into the LHS:
LHS = 1 - 1 / (sech^2(x))
11. Simplify the expression inside the parentheses:
LHS = 1 - sech^2(x)
12. Now, the LHS is equal to the right-hand side (RHS) of the equation.
Therefore, we have verified that 1 - tanh^2(x) = sech^2(x).