State the fundamental Pythagorean trigonometric identity for an angle \(\theta\).

If \(\sin\theta=\frac{3}{5}\) and \(\theta\) is acute, find \(\cos\theta\).

Use a Pythagorean identity to simplify \(1-\sin^2 x\).

Use a Pythagorean identity to simplify \(\sec^2 x-1\).

If \(\cos\theta=\frac{12}{13}\) and \(\theta\) is acute, find \(\sin\theta\).

Simplify \(\sin^2x+\cos^2x+\tan^2x\).

If \(\tan\theta=\frac34\) and \(\theta\) is in quadrant I, find \(\sec\theta\).

If \(\cot\theta=2\) and \(\theta\) is acute, find \(\csc\theta\).

Prove that \(\frac{1-\cos^2x}{\sin x}=\sin x\), assuming \(\sin x\ne0\).

Simplify \(\frac{\sec^2x-\tan^2x}{\cos x}\), where \(\cos x\ne0\).

Show that \((1-\sin x)(1+\sin x)=\cos^2x\).

A unit vector makes angle \(\theta\) with the positive \(x\)-axis. Its vertical component is \(\frac{7}{25}\), and it lies in quadrant I. Find its horizontal component and explain the identity used.

Simplify \(\frac{1}{1+\tan^2x}+\frac{1}{1+\cot^2x}\), where the expressions are defined.

If \(\sin\theta=-\frac{5}{13}\) and \(\theta\) is in quadrant III, find \(\tan\theta\).

Find all possible values of \(\cos\theta\) if \(\sin\theta=\frac{8}{17}\).

For which values of \(x\) is the simplification \(\frac{\sin^2x+\cos^2x}{\cos x}=\sec x\) valid?

A student says \(\sqrt{1-\sin^2x}=\cos x\) for every \(x\). Diagnose the error and give the correct statement.

Prove \(\frac{\tan^2x}{\sec x+1}=\sec x-1\), assuming both sides are defined.

Determine whether \(\frac{1}{1-\sin x}=\frac{1+\sin x}{\cos^2x}\) is an identity. State any restrictions.

Explain why the identity \(1+\tan^2\theta=\sec^2\theta\) is not valid at angles where \(\cos\theta=0\), even though it comes from \(\sin^2\theta+\cos^2\theta=1\).