Question

Assertion :lf $\tanh 5.\sinh \theta =\cosh 10-1$ then $\theta =10$ Reason: $\cosh 2x=\frac{1+{\tanh }^{2}x}{1-{\tanh }^{2}x}$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$tanh\left(5\right)sinh\left(10\right)=cosh\left(10\right)-1$
$sinh\left(5\right)sinh10=cosh\left(10\right)cosh\left(5\right)-cosh\left(5\right)$ using trignometric relations
$cosh\left(10\right)cosh\left(5\right)-sinh\left(5\right)sinh\left(10\right)=cosh\left(5\right)$
$cosh\left(5\right)=cosh\left(5\right)$
Hence LHS $=$ RHS.

Thus $\theta =10$ satisfies the above equation.

Also $\frac{1+ta{n}^{2}hx}{1-ta{n}^{2}hx}$
$=\frac{si{n}^{2}hx+co{s}^{2}hx}{co{s}^{2}hx-si{n}^{2}hx}$
$=coshxcoshx+sinhxsinhx$
$=cosh\left(2x\right)$
$=$LHS

Thus the assertion is well explained by the reason.