verify that the equation is an identity.\n\\frac{\\cos 2 x}{\\sin x}=\\cos x \\cot x-\\sin x\nto verify the…

verify that the equation is an identity.\n\\frac{\\cos 2 x}{\\sin x}=\\cos x \\cot x-\\sin x\nto verify the identity, start with either side and transform it to look like the other side. in this example, we will start with the right side. choose the correct transformations and transform the expression in\neach step.\n\\cos x \\cot x-\\sin x=\\square\n(do not simplify.)

verify that the equation is an identity.\n\\frac{\\cos 2 x}{\\sin x}=\\cos x \\cot x-\\sin x\nto verify the identity, start with either side and transform it to look like the other side. in this example, we will start with the right side. choose the correct transformations and transform the expression in\neach step.\n\\cos x \\cot x-\\sin x=\\square\n(do not simplify.)

Answer

Explanation:

Step1: Substitute cotangent identity

Recall that (\cot x=\frac{\cos x}{\sin x}). So, (\cos x\cot x-\sin x=\cos x\times\frac{\cos x}{\sin x}-\sin x).

Step2: Simplify the expression

[ \begin{align*} \cos x\times\frac{\cos x}{\sin x}-\sin x&=\frac{\cos^{2}x}{\sin x}-\sin x\ &=\frac{\cos^{2}x - \sin^{2}x}{\sin x} \end{align*} ]

Step3: Use double - angle formula

Recall the double - angle formula (\cos2x=\cos^{2}x-\sin^{2}x). So, (\frac{\cos^{2}x - \sin^{2}x}{\sin x}=\frac{\cos2x}{\sin x})

Answer:

(\frac{\cos2x}{\sin x})