Simplify and write the trigonometric expression in terms of sine and cosine:
tan^2 x-sec^2 x=
[(sin^2 x) / (cos^2 x)] - [1 / (cos^2 x)]
[(sin^2 x) - 1] / (cos^2 x)
-(cos^2 x) / (cos^2 x) = -1
To simplify the trigonometric expression tan^2x - sec^2x, let's start by writing the given trigonometric functions in terms of sine and cosine:
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)
To simplify and write the trigonometric expression tan^2(x) - sec^2(x) in terms of sine and cosine, we need to convert the tangent and secant functions into sine and cosine.
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.
since one of your basic trig identities is
sec^2 x = 1 + tan^2 x
this should not be too hard...