Using Pythagorean identities, state an equivalent expression.

\displaystyle \sin^2 \theta

Using Pythagorean identities, state an equivalent expression.

\displaystyle \tan^2 \theta

Using Pythagorean identities, state an equivalent expression.

\displaystyle \sec^2 \theta

Using Pythagorean identities, state an equivalent expression.

\displaystyle 1 - \sin^2 \theta

Using Pythagorean identities, state an equivalent expression.

\displaystyle 1 - \csc^2 \theta

Using Pythagorean identities, state an equivalent expression.

\displaystyle \cot^2\theta - \csc^2\theta

Using Pythagorean identities, state an equivalent expression.

\displaystyle -\frac{1}{\csc^2\theta}

Express each expression in a simpler form.

\displaystyle \cos \theta \sec \theta

Express each expression in a simpler form.

\displaystyle \tan \theta \cos \theta

Express each expression in a simpler form.

\displaystyle \tan \theta + \cot \theta

Express each expression in a simpler form.

\displaystyle \sqrt{1 -\cos^2\theta}

Express each expression in a simpler form.

\displaystyle \tan^2\theta -\sec^2 \theta

Express each expression in a simpler form.

\displaystyle \frac{\cos \theta}{1 + \sin \theta} + \frac{\cos \theta}{1 - \sin \theta}

Factor to simplify each expression.

\displaystyle \sin^4\theta +\sin^2\theta\cos^2 \theta

Factor to simplify each expression.

\displaystyle (\sec \theta)^2(\sin \theta)^2 + (\sin \theta)^2

Factor to simplify each expression.

\displaystyle 4\cos^2\theta + 8 \cos \theta \sin \theta + 4\sin^2 \theta

Factor to simplify each expression.

\displaystyle \sin^4\theta - \cos^4 \theta

Use the definitions of the primary trigonometric ratios in terms of x, y, and r to prove each identity.

\displaystyle \tan \theta \cos \tan \theta =\sin \tan \theta

Use the definitions of the primary trigonometric ratios in terms of x, y, and r to prove each identity.

\displaystyle \cot \theta \sec \theta = \csc \theta

Use the definitions of the primary trigonometric ratios in terms of x, y, and r to prove each identity.

\displaystyle \frac{1 + \cot^2\theta}{\csc^2\theta} = 1

Prove that \displaystyle \cos \theta \cot \theta = \frac{1}{\sin \theta} -\sin \theta

Prove that \displaystyle \frac{\cos^2\theta}{1 - \sin \theta} = 1 + \sin \theta

Prove that \displaystyle \frac{1 + \cot \theta}{\csc \theta} = \sin \theta + \cos \theta

Prove that \displaystyle \frac{\tan^2\theta}{1 + \tan^2\theta} = \sin^2\theta

Prove each identity.

\displaystyle \frac{\cos \theta \sin \theta}{\cot \theta} = 1- \cos^2\theta

Prove each identity.

\displaystyle \frac{\sin^2\theta}{1 - \cos \theta} = 1 + \cos \theta

Prove each identity.

\displaystyle \frac{1 + \sec \theta}{\sec \theta - 1} = \frac{1 + \cos \theta}{1 -\cos \theta}

Prove each identity.

\displaystyle \frac{1 -\sin \theta}{1 + \sin \theta} = \frac{\csc \theta -1}{\csc \theta + 1}

Simplify

\displaystyle \sin^2\theta + \frac{1 + \cot^2\theta}{1 + \tan^2\theta} + \cos^2\theta .

Prove that 2\sin(-\theta) - \cot\theta \sin\theta \cos\theta = (\sin \theta -1)^2 -2

Explain why the identity

\displaystyle \frac{\cos \theta}{1 + \sin \theta} + \frac{\cos \theta}{1 - \sin \theta} = \frac{2}{\cos \theta} is not true for \theta = 90^o and \theta= 270^o.

Prove that

\displaystyle \frac{1 + \sin \theta + \cos \theta}{1 -\sin \theta + \cos \theta} = \frac{1 + \sin \theta}{\cos \theta}

Prove that

\displaystyle \frac{1}{1 + \sin \theta} + \frac{1}{1- \sin \theta} = \frac{2}{\cos^2\theta} .

Prove that

\displaystyle \frac{7\sin \theta + 5\cos \theta}{\cos \theta \sin \theta} = 7 \sec \theta + 5\csc \theta .

Prove that

\displaystyle (\csc \theta - \cot \theta)^2 = \frac{1 -\cos \theta}{1 + \cos \theta} .

Prove that

\displaystyle \sec^2\theta + \csc^2\theta = \frac{\csc^2\theta}{\cos^2\theta} .

Prove that

\displaystyle \sec\theta - \frac{\sin \theta}{\cot \theta} = \frac{1}{\sec \theta} .

Prove that

\displaystyle \tan^2\theta \cos^2\theta = 1 - \frac{\sin^2 \theta}{\tan^2 \theta} .

Prove that

\displaystyle \frac{\tan \theta - \sin \theta}{\sin^3\theta} = \frac{\sec \theta}{1 + \cos \theta} .

Prove that

\displaystyle \frac{1 + \sin \theta}{\tan \theta} = \cos \theta + \cot \theta .

Prove that

\displaystyle (\tan \theta + \cot \theta)^2 = \sec^2\theta +\frac{1}{\sin^2\theta} .

Use a graphing calculator to graph each side to determine if the equation appears to be an identity.

\displaystyle \frac{1}{\sin^2x} + \frac{1}{\cos^2x} = 1

Use a graphing calculator to graph each side to determine if the equation appears to be an identity.

\displaystyle \cos x (\cos x -\sec x) = -\sin^2x

Prove that

\displaystyle (\sin \theta + \cos \theta)^2 = \frac{2 + \sec \theta \csc \theta}{\sec \theta \csc \theta}

Prove that

\displaystyle \frac{\cot \theta}{\csc \theta -1} = 2\sec \theta

Prove that

\displaystyle \frac{\tan \theta + \cos \theta}{\sin \theta} = \frac{1}{\cos \theta} + \frac{\cos \theta}{\sin \theta}

Prove that

\displaystyle \frac{\cos^2 \theta -\sin^2 \theta}{\cos^2 \theta + \sin \theta \cos \theta} = \frac{\cot \theta -1}{\cot \theta}

Prove that

\displaystyle \tan \theta \sin \theta + \cos \theta - \sec \theta + 1 = \sec^2\theta \cos^2 \theta

Prove that

\displaystyle \frac{1 -\tan^2\theta}{\tan \theta -\tan^2\theta} = 1 + \frac{1}{\tan \theta}

Prove that \tan^2\theta(1 + \cot^2\theta) = \sec^2\theta