Answer

Verified

428.7k+ views

**Hint**: Here we will use the formula for hyperbolic function and the formula of sine function expressed in exponential form. Since the standard formula is expressed in terms of “x”, we will convert the form in “ix” form. Then will simplify the equation for the resultant answer.

**:**

__Complete step-by-step answer__Using the identity for hyperbolic function –

$ \sinh x = \dfrac{{{e^x} - {e^{ - x}}}}{2} $

Here, the given function is with respect to therefore convert “x” to “ix” form

Therefore, the above equation can be expressed as –

$ \Rightarrow \sinh (ix) = \dfrac{{{e^{ix}} - {e^{ - ix}}}}{2} $ .... (A)

Also, $ {e^{ix}} = \cos x + i\sin x $

Place the above value in equation (A)

$ \Rightarrow \sinh (ix) = \dfrac{{(\cos x + i\sin x) - (\cos ( - x) + i\sin ( - x))}}{2} $

Now, use the properties of sines and cosines and apply accordingly. The odd and even trigonometric functions states that -

$

\sin ( - \theta ) = - \sin \theta \\

\cos ( - \theta ) = \cos \theta \;

$

$ \Rightarrow \sinh (ix) = \dfrac{{(\cos x + i\sin x) - (\cos x - i\sin x)}}{2} $

Now, open the brackets and simplify. Remember when there is a negative sign outside the bracket, all the terms inside the bracket changes. Positive term changes to negative and negative becomes positive.

$ \Rightarrow \sinh (ix) = \dfrac{{\cos x + i\sin x - \cos x + i\sin x}}{2} $

Make the pair of like terms and simplify –

$ \Rightarrow \sinh (ix) = \dfrac{{\underline {\cos x - \cos x} + \underline {i\sin x + i\sin x} }}{2} $

Like terms with equal value and opposite signs cancel each other. And add like terms with positive terms.

$ \Rightarrow \sinh (ix) = \dfrac{{2i\sin x}}{2} $

Common multiple from the numerator and the denominator cancel each other. Therefore remove from the numerator and the denominator.

$ \Rightarrow \sinh (ix) = i\sin x $

**So, the correct answer is “Option A”.**

**Note**: Always remember the standard hyperbolic formula for the trigonometric function and the exponential form of the trigonometric function. Remember the standard formula properly as the further solution depends on it. Be clear with the odd and even trigonometric function. Cosine is an even function whereas sine is an odd function.

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

A rainbow has circular shape because A The earth is class 11 physics CBSE

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

How do you graph the function fx 4x class 9 maths CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Why is there a time difference of about 5 hours between class 10 social science CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell