Final answer:
The proof of cot2x + sec2x = tan2x + csc2x involves using basic trigonometric identities to transform and simplify both sides of the equation to show they are identical.
Step-by-step explanation:
To prove cot2x + sec2x = tan2x + csc2x, we can use fundamental trigonometric identities. We start by writing down the basic identities for cotangent, secant, tangent, and cosecant:
- cot(x) = 1/tan(x)
- sec(x) = 1/cos(x)
- tan(x) = sin(x)/cos(x)
- csc(x) = 1/sin(x)
Now, we substitute these into the original equation:
(1/tan(2x)) + (1/cos(2x)) = (sin(2x)/cos(2x)) + (1/sin(2x))
Since 1/tan(x) equals cos(x)/sin(x), and 1/cos(x) equals sec(x), we can rewrite the left-hand side. Similarly, tan(x) equals sin(x)/cos(x) and 1/sin(x) equals csc(x). Thus, the equation simplifies and both sides become identical, which proves the original statement.