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Question: If \(\cos ec\theta = \coth x\), then the value of \(\tan \theta \) is: A. \(\cosh x\) B. \(\sin...

If cos⁡ecθ=coth⁡x\cos ec\theta = \coth x, then the value of tan⁡θ\tan \theta is:
A. cosh⁡x\cosh x
B. sinh⁡x\sinh x
C. tanh⁡x\tanh x
D. cos⁡echx\cos echx

Explanation

Solution

In the given problem, we are provided an equation involving the normal trigonometric functions and hyperbolic trigonometric functions. So, we have to find the value of the tangent function of an angle given the value of the cos\secant of the same angle in terms of hyperbolic trigonometric functions. We use the trigonometric formulae and identities such as tan⁡θ=1cot⁡θ\tan \theta = \dfrac{1}{{\cot \theta }} and cot⁡2θ+1=cos⁡ec2θ{\cot ^2}\theta + 1 = \cos e{c^2}\theta .

Complete step by step answer:
So, we have, cos⁡ecθ=coth⁡x\cos ec\theta = \coth x.Now, we have to find the value of tan⁡θ\tan \theta .
Using the trigonometric formula tan⁡θ=1cot⁡θ\tan \theta = \dfrac{1}{{\cot \theta }}, we get,
⇒tan⁡θ=1cot⁡θ\Rightarrow \tan \theta = \dfrac{1}{{\cot \theta }}
Now, we use the trigonometric formula cot⁡2θ+1=cos⁡ec2θ{\cot ^2}\theta + 1 = \cos e{c^2}\theta . So, we substitute the value of cot⁡θ\cot \theta in terms of cos⁡ecθ\cos ec\theta .
⇒tan⁡θ=1cos⁡ec2θ−1\Rightarrow \tan \theta = \dfrac{1}{{\sqrt {\cos e{c^2}\theta - 1} }}
Now, we have got the expression for tangent of angle θ\theta in terms of cosecant of the same angle. So, we know the value of cos⁡ecθ\cos ec\theta . Substituting the value in the expression, we get,
⇒tan⁡θ=1coth⁡2x−1\Rightarrow \tan \theta = \dfrac{1}{{\sqrt {{{\coth }^2}x - 1} }}

Now, we have the value of tan⁡θ\tan \theta in terms of hyperbolic trigonometric functions. Using the hyperbolic trigonometric identity coth⁡2x−cos⁡ech2x=1{\coth ^2}x - \cos ec{h^2}x = 1 in the expression, we get,
⇒tan⁡θ=1cos⁡ech2x\Rightarrow \tan \theta = \dfrac{1}{{\sqrt {\cos ec{h^2}x} }}
Computing the square root, we get,
⇒tan⁡θ=1cos⁡echx\Rightarrow \tan \theta = \dfrac{1}{{\cos echx}}
Now, using another hyperbolic trigonometric formula sinh⁡x=1cos⁡echx\sinh x = \dfrac{1}{{\cos echx}}. So, we get,
∴tan⁡θ=sinh⁡x\therefore \tan \theta = \sinh x
So, we get the value of tan⁡θ\tan \theta as sinh⁡x\sinh x.

So, option A is the correct answer.

Note: We must have knowledge about the hyperbolic trigonometric functions in order to solve the functions. One must know both the trigonometric and hyperbolic identities to get to the final answer. Take care of the calculations to be sure of the answer.We must use simplification rules to ease our calculations.