if sec x +tan x=p, then tan xeuqalto what. How?
To find the value of tan x in terms of p, we need to manipulate the given equation.
Let's start by squaring both sides of the equation:
(sec x + tan x)^2 = p^2
Expanding the left side of the equation using the identity (a + b)^2 = a^2 + 2ab + b^2, we get:
(sec^2 x + 2sec x tan x + tan^2 x) = p^2
Using the Pythagorean identity sec^2 x = 1 + tan^2 x, we can substitute sec^2 x with its equivalent form:
(1 + tan^2 x + 2sec x tan x + tan^2 x) = p^2
Combining like terms, we have:
2tan^2 x + 2sec x tan x + 1 = p^2
Rearranging the equation, we get:
2tan^2 x + 2sec x tan x = p^2 - 1
Factoring out a common factor of 2 from the left side of the equation:
2(tan^2 x + sec x tan x) = p^2 - 1
Dividing both sides of the equation by 2, we have:
tan^2 x + sec x tan x = (p^2 - 1)/2
Now, we can use the identity tan x = sin x / cos x and substitute it in the equation:
(sin x / cos x)^2 + sec x (sin x / cos x) = (p^2 - 1)/2
Expanding the equation by multiplying through by cos^2 x, we get:
sin^2 x + sec x sin x = (p^2 - 1) / (2cos^2 x)
Using the identity sec x = 1 / cos x, we can simplify it further:
sin^2 x + sin x / cos x = (p^2 - 1) / (2cos^2 x)
Multiplying through by cos x, we have:
sin x cos x + sin x = (p^2 - 1) / (2cos x)
Rearranging the terms, we get:
sin x (cos x + 1) = (p^2 - 1) / (2cos x)
Dividing both sides by cos x + 1, we have:
sin x = (p^2 - 1) / (2cos x (cos x + 1))
Finally, using the identity tan x = sin x / cos x, we substitute sin x back into the equation:
tan x = [(p^2 - 1) / (2cos x (cos x + 1))] / cos x
Simplifying further, we get:
tan x = (p^2 - 1) / (2cos^2 x + 2cos x)
Therefore, tan x is equal to (p^2 - 1) divided by (2cos^2 x + 2cos x).