Verify each trigonimetric equation by substituting identities to match right hand side of the equation to the hand side of the equation. Please help.
-tan^2x+sec^2x=1
sec^2= 1/cos^2, and -tan^2= -sin^2/cos^2 so on the left..
(-sin^2 +1)/cos^2=
(cos^2)/cos^2=
1
To verify the trigonometric equation -tan^2(x) + sec^2(x) = 1, we will substitute the identities for tangent and secant functions to match the right-hand side of the equation with the left-hand side.
1. Start with the left-hand side of the equation: -tan^2(x) + sec^2(x)
2. Substitute the identity for the tangent function:
Since tan(x) = sin(x)/cos(x), we can rewrite -tan^2(x) as -sin^2(x)/cos^2(x).
Our equation now becomes: -sin^2(x)/cos^2(x) + sec^2(x)
3. Substitute the identity for the secant function:
Since sec(x) = 1/cos(x), we can rewrite sec^2(x) as (1/cos(x))^2, which simplifies to 1/cos^2(x).
Our equation now becomes: -sin^2(x)/cos^2(x) + 1/cos^2(x)
4. To combine the two terms, we need a common denominator. Since both terms have a denominator of cos^2(x), the common denominator is cos^2(x).
Our equation now becomes: (-sin^2(x) + 1) / cos^2(x)
5. Simplify the numerator by using the identity sin^2(x) + cos^2(x) = 1:
(-sin^2(x) + 1) is equivalent to (-1 * cos^2(x) + 1) = 1 - cos^2(x).
Our equation now becomes: (1 - cos^2(x)) / cos^2(x)
6. Simplify further:
(1 - cos^2(x)) / cos^2(x) can be rewritten as sin^2(x) / cos^2(x) by using the identity sin^2(x) = 1 - cos^2(x).
Our equation now becomes: sin^2(x) / cos^2(x)
7. Finally, using the identity tan^2(x) = sin^2(x) / cos^2(x), we can rewrite the right-hand side of the equation:
sin^2(x) / cos^2(x) is equivalent to tan^2(x).
Our equation now becomes: tan^2(x)
8. Since our final result on the right-hand side matches the left-hand side of the equation, we have verified that -tan^2(x) + sec^2(x) = 1.
Therefore, the trigonometric equation -tan^2(x) + sec^2(x) = 1 is true.