Question

If \[7\sin \alpha = 24\cos \alpha \] where \[0 < \alpha < \dfrac{\pi }{2}\] , then value of \[14\tan \alpha - 75\cos \alpha - 7\sec \alpha \] is equals to
A) \[1\]
B) \[2\]
C) \[3\]
D) \[4\]

Answer 1

Hint: We know that, the trigonometric functions like secant and tangent are defined as follows,
\[
  \sec \alpha = \dfrac{1}{{\cos \alpha }} \\
  \tan \alpha = \dfrac{{\sin \alpha }}{{\cos \alpha }} \\
 \]
Relation between \[\tan \alpha ,\sec \alpha \] is by formula
\[{\sec ^2}\alpha - {\tan ^2}\alpha = 1\]

Complete step-by-step answer:
It is given in question that, \[7\sin \alpha = 24\cos \alpha \] where \[0 < \alpha < \dfrac{\pi }{2}\]
Now consider the given equation,
\[7\sin \alpha = 24\cos \alpha \]
Let us divide both sides of the above function by \[7\cos \alpha \], then we get,
\[\dfrac{{\sin \alpha }}{{\cos \alpha }} = \dfrac{{24}}{7}\]
We know by trigonometric identities that\[\dfrac{{\sin \alpha }}{{\cos \alpha }} = \tan \alpha \],
On substituting the above identity in the equation we get,
\[\tan \alpha = \dfrac{{24}}{7}\]
Let us apply the value of \[\tan \alpha \]in one of the trigonometric identity which is given below, we get,
\[{\sec ^2}\alpha - {\tan ^2}\alpha = 1\]
On applying we get,
\[{\sec ^2}\alpha - {(\dfrac{{24}}{7})^2} = 1\]
Let us rearrange and solve the above equation, after rearranging we get,
\[{\sec ^2}\alpha = 1 + {(\dfrac{{24}}{7})^2}\]
Now we shall start solving the equation,
\[
  {\sec ^2}\alpha = \dfrac{{49 + 576}}{{49}} \\
  {\sec ^2}\alpha = \dfrac{{625}}{{49}} \\
 \]
Now let us take square root on both sides then we get,
\[
  \sec \alpha = \sqrt {\dfrac{{625}}{{49}}} \\
  \sec \alpha = \dfrac{{25}}{7} \\
 \]
As we know that \[\sec \alpha = \dfrac{1}{{\cos \alpha }}\]and the vice versa also holds. We will find the value of\[\cos \alpha \] with the vice versa of the said identity,
Now we get\[\cos \alpha = \dfrac{7}{{25}}\].
Now we have got the following values
\[\cos \alpha = \dfrac{7}{{25}}\],\[\sec \alpha = \dfrac{{25}}{7}\]and\[\tan \alpha = \dfrac{{24}}{7}\].
Now let us substitute the values found in the equation to which we have to find the solution,
We are in need to find the value of
\[14\tan \alpha - 75\cos \alpha - 7\sec \alpha \]
Now let us substitute the values, we get
\[14\tan \alpha - 75\cos \alpha - 7\sec \alpha = 14 \times \dfrac{{24}}{7} - 75 \times \dfrac{7}{{25}} - 7 \times \dfrac{{25}}{7}\]
On simplifying the above equation we get,
\[14\tan \alpha - 75\cos \alpha - 7\sec \alpha = 48 - 21 - 25 = 2\]
Hence we have found the value of \[14\tan \alpha - 75\cos \alpha - 7\sec \alpha \] as 2.

Hence we can come to a conclusion that option (B) is the correct option.

Note:
\[{\sec ^2}\alpha - {\tan ^2}\alpha = 1\]is the formula used in the problem. Similarly we have many trigonometric identities some are listed below,
$
  {\sin ^2}x + {\cos ^2}x = 1 \\
  \dfrac{1}{{\sin x}} = \cos ecx \\
  \dfrac{1}{{\tan x}} = \cot x \\
 $