Question

for each expression, choose the expression that completes an identity. one or both expressions may need to be rewritten. fill in each blank with the appropriate expression. - cot x sin x = sec²x - 1 = sec x / csc x = 1 + sin²x = cos²x = - cos x sin²x / cos²x tan x csc²x - cot²x + sin²x 1 / sec²x

Explanation:

Step1: Simplify \(-\cot x\sin x\)

Use the identity \(\cot x=\frac{\cos x}{\sin x}\). Then \(-\cot x\sin x =-\frac{\cos x}{\sin x}\cdot\sin x\).

Step2: Cancel out \(\sin x\)

After canceling \(\sin x\) in \(-\frac{\cos x}{\sin x}\cdot\sin x\), we get \(-\cos x\).

Step3: Simplify \(\sec^{2}x - 1\)

Use the Pythagorean identity \(\sec^{2}x=1 + \tan^{2}x\). So \(\sec^{2}x-1=\tan^{2}x=\frac{\sin^{2}x}{\cos^{2}x}\).

Step4: Simplify \(\frac{\sec x}{\csc x}\)

Use the identities \(\sec x=\frac{1}{\cos x}\) and \(\csc x=\frac{1}{\sin x}\). Then \(\frac{\sec x}{\csc x}=\frac{\frac{1}{\cos x}}{\frac{1}{\sin x}}=\frac{\sin x}{\cos x}=\tan x\).

Step5: Simplify \(1+\sin^{2}x\)

Use the Pythagorean identity \(\csc^{2}x-\cot^{2}x = 1\). So \(1+\sin^{2}x=\csc^{2}x-\cot^{2}x+\sin^{2}x\).

Step6: Simplify \(\cos^{2}x\)

Use the identity \(\cos^{2}x=\frac{1}{\sec^{2}x}\).

Answer:

\(-\cot x\sin x=-\cos x\)

\(\sec^{2}x - 1=\frac{\sin^{2}x}{\cos^{2}x}\)

\(\frac{\sec x}{\csc x}=\tan x\)

\(1+\sin^{2}x=\csc^{2}x-\cot^{2}x+\sin^{2}x\)

\(\cos^{2}x=\frac{1}{\sec^{2}x}\)