Is the given equation an identity or a conditional eqution
sin(-x) tan(-x) + cos(-x) = sec x
let's get rid of all the - signs
sin(x) tan(x) + cos(x) = sec x
sin^2x/cosx + cosx = secx
(sin^2x + cos^2x)/cosx = secx
now what do you think?
To determine whether the given equation is an identity or a conditional equation, we need to simplify both sides of the equation separately and then compare them.
First, let's simplify the left side of the equation:
sin(-x) tan(-x) + cos(-x)
Using trigonometric identity, we can rewrite sin(-x) and cos(-x) in terms of sin(x) and cos(x):
sin(-x) = -sin(x)
cos(-x) = cos(x)
Let's substitute these values back into the equation:
-sin(x) * tan(-x) + cos(x)
Next, using another trigonometric identity, we can rewrite tan(-x) in terms of tan(x):
tan(-x) = -tan(x)
Substituting this value back into the equation:
-sin(x) * (-tan(x)) + cos(x)
Now, simplify further:
sin(x) * tan(x) + cos(x)
Using the definition of tangent (tan(x) = sin(x)/cos(x)), we can rewrite the equation as:
sin(x) * (sin(x) / cos(x)) + cos(x)
Simplifying the expression:
(sin^2(x) / cos(x)) + cos(x)
Now, using the definition of secant (sec(x) = 1/cos(x)), we can rewrite the equation as:
sin^2(x) / cos(x) + cos(x) = sec(x)
Thus, we have shown that the left side of the equation is equal to sec(x), which is also the right side of the equation.
Since both sides of the equation are equal, we can conclude that the given equation is an identity.