Question

If $\tan \left( {\pi \cos \theta } \right)\, = \,\cot \left( {\pi \sin \theta } \right)$ then what is the value of $19208{\cos ^2}\left( {\theta - \pi /4} \right)\,$is equal to

Answer 1

There are one or more trigonometric ratios to these equations, we find the value of such a function in terms of an angle, it is expressed in a simplified form. The Trigonometric Identities for Right Angled Triangles are true equations. By using as a reference, a right-angled triangle, the trigonometric functions

Formula used:
Cofunctions identities, formula
According to the trigonometric function,
$\tan \left( {\dfrac{\pi }{2} - \theta } \right) = \cot \theta$
$\cos \left( {\dfrac{\pi }{2} - \theta } \right) = \sin \theta$
$\cos \left( {\dfrac{\pi }{4} - \theta } \right) = \dfrac{{2n + 1}}{{2\sqrt 2 }}$
Since $- 1 < \cos \left( {\dfrac{\pi }{4} - \theta } \right) < 1$
Where,
$\sin \theta ,\tan \theta ,\cos \theta$are the trigonometric identities,
$\theta$ representing the angle,
$\dfrac{\pi }{2}$ is an angle of ${90^ \circ }$

Complete step-by-step answer:
Given by,
$\tan \left( {\pi \cos \theta } \right)\, = \,\cot \left( {\pi \sin \theta } \right)$
Find the value of $19208{\cos ^2}\left( {\theta - \pi /4} \right)\,$
In trigonometric identities,
We know that,
$\tan \left( {\dfrac{\pi }{2} - \theta } \right) = \cot \theta$
Substituting the given value, we get,
$\Rightarrow$ $\theta = \pi \cos \theta$ in above equation,
$\Rightarrow$ $\tan \left( {\pi \cos \theta } \right) = \tan \left( {\dfrac{\pi }{2} - \pi \sin \theta } \right)$
The range of an identities is given below,
$\Rightarrow$ $\pi \cos \theta = n\pi + \dfrac{\pi }{2} - \pi \sin \theta \left( {n \in I} \right)$
Now, we rearranging the above equation is,
Here,
$\Rightarrow$ $\pi \left( {\sin \theta + \cos \theta } \right) = \left( {2n + 1} \right)\dfrac{\pi }{2}$
Both sides the $\pi$ canceled,
$\Rightarrow$ $\sin \theta + \cos \theta = \dfrac{{2n + 1}}{2}$
The trigonometric identities is
$\Rightarrow$ $\cos \left( {\dfrac{\pi }{2} - \theta } \right) = \sin \theta$
So, we can simplify the equation
$\Rightarrow$ $\sin \theta + \cos \theta = \dfrac{{2n + 1}}{2}$
Similarly, we get,
$\Rightarrow$ $\cos \left( {\dfrac{\pi }{4} - \theta } \right) = \dfrac{{2n + 1}}{{2\sqrt 2 }}$………………..$\left( 1 \right)$
The range between $- 1,1$
Since $- 1 < \cos \left( {\dfrac{\pi }{4} - \theta } \right) < 1$
Substituting the above value in range
$\Rightarrow$ $- 1 < \dfrac{{2n - 1}}{{2\sqrt 2 }} < 1$
Substituting the value $n = 0, - 1$
Where, $n$ is an integer
$\Rightarrow$ $\cos \left( {\dfrac{\pi }{4} - \theta } \right) = \pm \left( {\dfrac{1}{{2\sqrt 2 }}} \right)$
Rearranging the above equation
We get,
Let $2$ can be written as $\sqrt 2 \sqrt 2$
So, $2 \times 2 \times 2{\cos ^2}\left( {\pi /4 - \theta } \right) = 1$
On simplifying,
$8{\cos ^2}\left( {\pi /4 - \theta } \right) = 1$
Now we find the value of $19208{\cos ^2}\left( {\theta - \pi /4} \right)\,$
Here, divide by $8$in $19208$
We get,$2401$
On simplifying a given equation,
$\Rightarrow$ $19208{\cos ^2}\left( {\theta - \pi /4} \right)\,$ $= 2401$
Hence, thus, the value of $19208{\cos ^2}\left( {\theta - \pi /4} \right)\,$is equal to $2401$

Note: These equations have unknown angles for one or more trigonometric ratios. If you are not so sure about the values of different angles, the trigonometry table is a table you may refer to Sine, cosine, tangent, cotangent, cosecant, and secant trigonometric functions that describe the interaction between the triangle's sides and angles.