To express tan x in terms of sec x, we start from the relationships in trigonometry.
We know that:
- tan x = sin x / cos x
- sec x = 1 / cos x
From the definition of sec x, we can express cos x in terms of sec x:
cos x = 1 / sec x
Now, substituting this expression into the formula for tan x, we have:
tan x = sin x / (1 / sec x)
This simplifies to:
tan x = sin x * sec x
Next, we need to express sin x in terms of sec x. Using the Pythagorean identity, we know:
sin² x + cos² x = 1
Substituting cos x = 1 / sec x into the identity gives:
sin² x + (1 / sec x)² = 1
Solving for sin² x gives:
sin² x = 1 – (1 / sec² x)
Thus, we have:
sin² x = (sec² x – 1) / sec² x
Taking the square root, we find:
sin x = √((sec² x – 1) / sec² x) = √(sec² x – 1) / sec x
Now substituting this back into our expression for tan x:
tan x = (√(sec² x – 1) / sec x) * sec x
Finally, we simplify to:
tan x = √(sec² x – 1)
In conclusion, we can express tan x in terms of sec x as:
tan x = √(sec² x – 1)