Question

If $ {\tanh ^2}x = {\tan ^2}\theta , $ then $ \cosh 2x = $
 $
  1) - \sin 2\theta \\
  2)\sec 2\theta \\
  3)\cos 3\theta \\
  4)\cos 2\theta \\
  $

Answer 1

Hint: Here, we will use different identities for hyperbolic cosine and the tangent function and find the correlations between the given and the required term and simplify for the required resultant value.

Complete step-by-step answer:
Take the given expression: $ \cosh 2x $
We know the identity that: $ \cosh 2x = \dfrac{{1 + {{\tanh }^2}x}}{{1 - {{\tanh }^2}x}} $
Place the given value $ {\tanh ^2}x = {\tan ^2}\theta , $ in the above expression –
 $ \cosh 2x = \dfrac{{1 + {{\tan }^2}\theta }}{{1 - {{\tan }^2}\theta }} $
The above expression can be re-written as –
 $ \cosh 2x = \dfrac{1}{{\dfrac{{1 - {{\tan }^2}\theta }}{{1 + {{\tan }^2}\theta }}}} $ (As the denominator’s denominator goes to the numerator and vice-versa)
Now, use the identity - $ \dfrac{{1 - {{\tan }^2}\theta }}{{1 + {{\tan }^2}\theta }} = \cos 2\theta $ place the value in the above equation –
 $ \cosh 2x = \dfrac{1}{{\cos 2\theta }} $
Now the reciprocal of the cosine function is the secant function, replace the above expression using it.
 $ \cosh 2x = \sec 2\theta $
Hence, from the given multiple choices – the second option is the correct answer.
So, the correct answer is “Option 2”.

Note: Always remember different trigonometric identities for the angles and the hyperbolic angles and place carefully. Know the reciprocal of the trigonometric functions such as sine and cosine are reciprocal of each other. Similarly, tangent and cotant, secant and cosec are the inverses of each other.