Question

Find the value of $[\tan A /(1+\sec A)]+[(1+\sec A) / \tan A]$
1) $2 \sin A$
2) $2 \cos \mathrm{A}$
3) $2 \operatorname{cosec} \mathrm{A}$
4) $2 \sec A$

Answer 1

Hint: We have to find the value of the given expression for that we need to know some basic trigonometric theorems. All trigonometric identities are built upon the foundation of the six trigonometric ratios. Sine, cosine, tangent, cosecant, secant, and cotangent are a few of their names. These trigonometric ratios are each defined in terms of the right triangle's adjacent side, opposite side, and hyperbolic tangent side. All basic trigonometric identities are derived from the six trigonometric ratios.

Formula Used:
The trignometric identity ${{\sec }^{2}}x-{{\tan }^{2}}x=1$

Complete step by step Solution:
Given that
${[\tan A /(1+\sec A)]+[(1+\sec A) / \tan A] }$
Cross-multiply the terms
We get
$=\left[\left(\tan ^{2} A+(1+\sec A)^{2}\right] /(1+\sec A) \tan A\right.$
There are two primary tangent function formulations. As we already know, $\tan x$ in a right-angled triangle is defined as the ratio of the angle's opposite and adjacent sides. The ratio of the sine function to the cosine function, which can be calculated using a unit circle, is another way to define the tangent function. So, the $\tan x$ formulae are as follows:.
- $\tan x=\sin x / \cos x$
- $\tan x=$ Opposite Side/Adjacent Side
using the identity ${{\sec }^{2}}x-{{\tan }^{2}}x=1$
$=\left[ \left( {{\sec }^{2}}A-1 \right)+{{(1+\sec A)}^{2}} \right]/(1+\sec A)\tan A$
$=[(\sec A+1)(\sec A-1+1+\sec A)] /(1+\sec A) \tan A$
$=2 \sec A / \tan A$
$=(2 / \cos A) /(\sin A / \cos A)$
$=2 / \sin A$
$=2 \operatorname{cosec} A$

Hence, the correct option is 3.

Additional Information:There are two primary tangent function formulations. Since we already know, in a right-angled triangle is defined as the ratio of the angle's opposite and adjacent sides. The ratio of the sine function to the cosine function, which can be calculated using a unit circle, is another way to define the tangent function.

Note: It's important to keep in mind that the cosecant is the relationship between two sides of a right-angled triangle and an acute angle. Any angle that is less than or greater than 90 degrees can have a cosecant since both acute and obtuse angles can have a cosecant.