prove the identity. the numbered boxes correspond to the answer box. parentheses, it is a hint to you that…

prove the identity. the numbered boxes correspond to the answer box. parentheses, it is a hint to you that the expression that belongs in it is eitl sure there are no spaces in your answers. tan x = csc 2x - cot 2x right side = 1/1 - 2/1 = 1 - 2/1 = 1 - (3)/4 = 5/4 = sin x/cos x = tan x 1 = 2 = 3 = 4 = 5 =

prove the identity. the numbered boxes correspond to the answer box. parentheses, it is a hint to you that the expression that belongs in it is eitl sure there are no spaces in your answers. tan x = csc 2x - cot 2x right side = 1/1 - 2/1 = 1 - 2/1 = 1 - (3)/4 = 5/4 = sin x/cos x = tan x 1 = 2 = 3 = 4 = 5 =

Answer

Explanation:

Step1: Recall reciprocal and quotient identities

We know that $\csc2x=\frac{1}{\sin2x}$ and $\cot2x = \frac{\cos2x}{\sin2x}$. So, $\csc2x-\cot2x=\frac{1}{\sin2x}-\frac{\cos2x}{\sin2x}=\frac{1 - \cos2x}{\sin2x}$.

Step2: Use double - angle formulas

The double - angle formulas are $\cos2x=1 - 2\sin^{2}x$ and $\sin2x = 2\sin x\cos x$. Substituting $\cos2x = 1 - 2\sin^{2}x$ into $\frac{1-\cos2x}{\sin2x}$, we get $\frac{1-(1 - 2\sin^{2}x)}{2\sin x\cos x}$.

Step3: Simplify the expression

Simplify $\frac{1-(1 - 2\sin^{2}x)}{2\sin x\cos x}=\frac{2\sin^{2}x}{2\sin x\cos x}=\frac{\sin x}{\cos x}=\tan x$.

  1. $\sin2x$
  2. $\cos2x$
  3. $1 - 2\sin^{2}x$
  4. $2\sin x\cos x$
  5. $2\sin^{2}x$

Answer:

  1. $\sin2x$
  2. $\cos2x$
  3. $1 - 2\sin^{2}x$
  4. $2\sin x\cos x$
  5. $2\sin^{2}x$