Verify the identity.
\[ \sec{x} - \cos{x} = \sin{x} \cdot \tan{x} \]
Verify the identity.
\[ \frac{1+\csc{\beta}}{\sec{\beta}} - \cot{\beta} = \cos{\beta} \]
Verify the identity.
\[ \frac{\cos{y} \cdot \cot{y}}{1-\sin{y}}-1=\csc{y} \]
Verify the identity.
\[ \frac{\csc{N}+1}{\cot{N}} = \frac{ \cot{N} }{ \csc{N} -1} \]
Verify the identity.
\[ \sqrt{\frac{1+\sin{x}}{1-\sin{x}}} = \frac{1+ \sin{x}}{ | \cos{x} | } \]
Verify the identity.
\[ \sin \bigg({\frac{\pi}{2}-x} \bigg)=\cos{x} \]
Verify the identity.
\[ \cos \bigg({\frac{\pi}{2}-x} \bigg)=\sin{x} \]
Verify the identity.
\[ \sin({\pi-x})=\sin{x} \]
Write the trigonometric expression as an algebraic expression.
\[ \sin( \arccos{x} - \arcsin{x} ) \]
Write the trigonometric expression as an algebraic expression.
\[ \cos( \arctan{x} - \arcsin{x} ) \]
Write the trigonometric expression as an algebraic expression.
\[ \cos \Bigg( \tan^{-1}{\sqrt{x+3}} - \tan^{-1}{\frac{\sqrt{x}}{2}} \Bigg) \]
Angles \( x \) and \( y \) are in Quadrant II.
\[ \sin{x}=\frac{12}{13} \quad \quad \cos{y}=-\frac{3}{5} \] Find: \[ \sin{(x+y)} \quad \quad \cot{(x-y)}\]
Angles \( a \) and \( b \) are in Quadrant III.
\[ \tan{a}=\frac{1}{3} \quad \quad \sec{b}=-\frac{5}{4} \] Find: \[ \cos{(a-b)} \]
Angles \( a \) and \( b \) are in Quadrant IV.
\[ \csc{a}=-\frac{3\sqrt{6}}{2} \quad \quad \cos{b}=\frac{\sqrt{7}}{4} \] Find: \[ \tan{(a+b)} \]
Angle \( \theta \) is in Quadrant I.
\[ \tan{\theta}=\frac{5}{12} \] Find: \[ \sin{(2\theta)} \quad \quad \sec{(2\theta)}\]
Angle \( \theta \) is in Quadrant I.
\[ \csc{\theta}=\sqrt{10} \] Find: \[ \cos{(2\theta)} \quad \quad \cot{(2\theta)}\]
Angle \( \theta \) is in Quadrant I.
\[ \sin{\theta}=\frac{1}{3} \] Find: \[ \tan{(2\theta)} \quad \quad \csc{(2\theta)}\]
Verify the given \(x\)-values are solutions of the equation.
\[ 4 \sin^2{(4x)}-3=0 \] \[ x=\frac{\pi}{12}, \frac{5 \pi}{6} \]
Verify the given \(x\)-values are solutions of the equation.
\[ \csc{\Big( \frac{x}{3} \Big) }-2=0 \] \[ x=-\frac{7 \pi}{2}, \frac{\pi}{2} \]
Verify the given \(x\)-values are solutions of the equation.
\[ \tan^2{x}-\tan{x}-2=0 \] \[ x=\frac{3 \pi}{4}, \tan^{-1}{(2)} \]
Solve in the interval \( [0, 2\pi)\).
\[ \cos{x} + \sqrt{3} = - \cos{x} \]
Solve in the interval \( [0, 2\pi)\).\[ 3\tan^{2}{x}-1=0 \]
Solve in the interval \( [0, 2\pi)\).\[ 2\cos^{2}{x}+\cos{x}-1=0 \]
Solve in the interval \( [0, 2\pi)\).\[ \sin{x}-1=\cos{x} \]
Solve in the interval \( [0, 2\pi)\).\[ 5\tan^{2}{(2x)}-15=0 \]
Solve in the interval \( [0, 2\pi)\).\[ 2\sin{ \Big( \frac{x}{3} \Big) }+\sqrt{3}=0 \]
Solve in the interval \( [0, 2\pi)\).\[ \sin \Big( x+\frac{\pi}{3} \Big) +\sin \Big( x-\frac{\pi}{3} \Big) =-1 \]
Solve in the interval \( [0, 2\pi)\).\[ \cos \Big( x+\frac{\pi}{6} \Big) -\cos \Big( x-\frac{\pi}{6} \Big) =-1 \]
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