tan^2 x-sec^2 x=
R_scott
answered
[(sin^2 x) - 1] / (cos^2 x)
-(cos^2 x) / (cos^2 x) = -1
oobleck
answered
sec^2 x = 1 + tan^2 x
this should not be too hard...
Step-by-Step Bot
answered
tan(x) = sin(x) / cos(x)
sec(x) = 1 / cos(x)
Now, let's substitute these values back into the original expression:
tan^2(x) - sec^2(x) = (sin(x) / cos(x))^2 - (1 / cos(x))^2
Next, simplify the expression by expanding the squares:
(sin(x) / cos(x))^2 = sin^2(x) / cos^2(x)
(1 / cos(x))^2 = 1 / cos^2(x)
Substituting these back into the original expression:
tan^2(x) - sec^2(x) = sin^2(x) / cos^2(x) - 1 / cos^2(x)
To combine these fractions, we need a common denominator, which is cos^2(x):
tan^2(x) - sec^2(x) = (sin^2(x) - 1) / cos^2(x)
As a final step, we can recognize that sin^2(x) - 1 is the identity for cos^2(x):
sin^2(x) - 1 = cos^2(x) - 1
Therefore, the simplified expression is:
tan^2(x) - sec^2(x) = (cos^2(x) - 1) / cos^2(x)
Explain Bot
answered
We'll start by rewriting the expression using the definitions of tangent and secant:
tan^2(x) - sec^2(x) = (sin^2(x)/cos^2(x)) - (1/cos^2(x))
Now, let's find a common denominator to combine the fractions:
tan^2(x) - sec^2(x) = (sin^2(x) - 1) / cos^2(x)
Next, we'll use the Pythagorean Identity for sine and cosine:
sin^2(x) - 1 = 1 - cos^2(x) - 1 = -cos^2(x)
Now, we can substitute this into the expression:
tan^2(x) - sec^2(x) = -cos^2(x) / cos^2(x)
Finally, simplifying further:
tan^2(x) - sec^2(x) = -1
Therefore, the simplified expression tan^2(x) - sec^2(x) = -1 in terms of sine and cosine.
