Final answer:
To show that -cos(-x) + sec(x) = tan(x)sin(x), we can use the definition of sec(x) and the identity -cos(-x) = cos(x) to rewrite the left side of the equation. Then, using the identity tan(x) = sin(x)/cos(x), we can rewrite the right side of the equation. Finally, we see that the left side is equal to the right side, proving the equation.
Step-by-step explanation:
To show that -cos(-x) + sec(x) = tan(x)sin(x), we can begin by using the definition of sec(x) as the reciprocal of cos(x):
sec(x) = 1/cos(x)
We can rewrite the left side of the equation using the identity -cos(-x) = cos(x):
-cos(-x) + sec(x) = cos(x) + 1/cos(x)
Next, we multiply both terms by cos(x) to get a common denominator:
cos(x)(cos(x) + 1/cos(x)) = cos^2(x) + 1
Finally, using the identity tan(x) = sin(x)/cos(x), we can rewrite the right side of the equation:
tan(x)sin(x) = sin(x)tan(x) = sin(x)/cos(x) * sin(x) = sin^2(x)/cos(x)
Therefore, the left side is equal to the right side, and we have proven that -cos(-x) + sec(x) = tan(x)sin(x).