Trigonometric Identities

Trigonometric identities are statements about the relationships of trigonometric ratios. These include reciprocal ratios, complementary angles and the rules for the angle of any magnitude.

Several relationships are shown below:

Reciprocal Ratio & Angles of Any Magnitude: $\csc { \theta } =\cfrac { 1 }{ \sin { \theta } } \\ \\ \\ \\ \sec { \theta } =\cfrac { 1 }{ \cos { \theta } } \\ \\ \\ \\ \cot { \theta } =\cfrac { 1 }{ \tan { \theta } }\\ \\ \\ \\ \sin { \theta } =\cos { (90^{ \circ }-\theta ) } \\ \csc { \theta } =\sec { (90^{ \circ }-\theta ) } \\ \tan { \theta } =\cot { (90^{ \circ }-\theta ) }$ $\sin { (180^{ \circ }-\theta ) } =\sin { \theta } \\ \cos { (180^{ \circ }-\theta ) } =-\cos { \theta } \\ \tan { (180^{ \circ }-\theta ) } =-\tan { \theta } \\ \\ \\ \\ \sin { (180^{ \circ }+\theta ) } =-\sin { \theta } \\ \cos { (180^{ \circ }+\theta ) } =-\cos { \theta } \\ \tan { (180^{ \circ }+\theta ) } =\tan { \theta } \\ \\ \\ \\ \sin { (360^{ \circ }-\theta ) } =-\sin { \theta } \\ \cos { (360^{ \circ }-\theta ) } =\cos { \theta } \\ \tan { (360^{ \circ }-\theta ) } =-\tan { \theta } \\ \\ \\ \\ \sin { (-\theta ) } =-\sin { \theta } \\ \cos { (-\theta ) } =\cos { \theta } \\ \tan { (-\theta ) } =-\tan { \theta }$ Examples:

1. Simplify $\sin { \theta } cot\theta$. $\sin { \theta } cot\theta =\sin { \theta } \times \cfrac { \cos { \theta } }{ \sin { \theta } } \\ \\ \qquad \qquad \quad =\cos { \theta }$

2. Simplify $\sin { (90^\circ-\beta ) } \sec { \beta }$ where $\beta$ is an acute angle. $\sin { (90^\circ-\beta ) } \sec { \beta } =\cos { \beta } \times \cfrac { 1 }{ \cos { \beta } } \\ \qquad \qquad \qquad \qquad = 1$

3. Simplify $\sqrt { \sin ^{ 4 }{ \theta } +\sin ^{ 2 }{ \theta } \cos ^{ 2 }{ \theta } }$ $\sqrt { \sin ^{ 4 }{ \theta } +\sin ^{ 2 }{ \theta } \cos ^{ 2 }{ \theta } } =\sqrt { \sin ^{ 2 }{ \theta } (\sin ^{ 2 }{ \theta } +\cos ^{ 2 }{ \theta } ) } \\ \qquad \qquad \qquad \qquad \qquad \quad \quad =\sqrt { \sin ^{ 2 }{ \theta } (1) } \\ \qquad \qquad \qquad \qquad \qquad \quad \quad =\sqrt { \sin ^{ 2 }{ \theta } } \\ \qquad \qquad \qquad \qquad \qquad \quad \quad =\sin { \theta }$

4. Prove $\cot { x } +\tan { x } =\csc { x } \sec { x }$. $\cot { x } +\tan { x } =\csc { x } \sec { x } \\ \\ LHS=\cot { x } +\tan { x } \\ \\ \qquad =\cfrac { \cos { x } }{ \sin { x } } +\cfrac { \sin { x } }{ \cos { x } } \\ \\ \qquad =\cfrac { \cos ^{ 2 }{ x } +\sin ^{ 2 }{ x } }{ \sin { x } \cos { x } } \\ \\ \qquad =\cfrac { 1 }{ \sin { x } \cos { x } } \\ \\ \qquad =\cfrac { 1 }{ \sin { x } } \times \cfrac { 1 }{ \cos { x } } \\ \\ \qquad =\csc { x } \sec { x } \\ \\ \qquad =RHS$

5. Prove that $\cfrac { 1-\cos { x } }{ \sin ^{ 2 }{ x } } =\cfrac { 1 }{ 1+\cos { x } }$. $\cfrac { 1-\cos { x } }{ \sin ^{ 2 }{ x } } =\cfrac { 1 }{ 1+\cos { x } } \\ \\ LHS=\cfrac { 1-\cos { x } }{ \sin ^{ 2 }{ x } } \\ \\ \qquad =\cfrac { 1-\cos { x } }{ 1-\cos ^{ 2 }{ x } } \\ \\ \qquad =\cfrac { 1-\cos { x } }{ (1+\cos { x } )(1-\cos { x } ) } \\ \\ \qquad =\cfrac { 1 }{ 1+\cos { x } } \\ \\ \qquad =RHS$

6. Find the exact value of $\cos { 75 }$. $\cos { 75 } =\cos { (30+45) } \\ \qquad =\cos { 30 } \cos { 45 } -\sin { 30 } \sin { 45 } \\ \\ \qquad =\cfrac { \sqrt { 3 } }{ 2 } \times \cfrac { 1 }{ \sqrt { 2 } } -\cfrac { 1 }{ 2 } \times \cfrac { 1 }{ \sqrt { 2 } } \\ \\ \qquad =\cfrac { \sqrt { 3 } -1 }{ 2\sqrt { 2 } } \\ \\ \qquad =\cfrac { \sqrt { 3 } -1 }{ 2\sqrt { 2 } } \times \cfrac { \sqrt { 2 } }{ \sqrt { 2 } } \\ \\ \qquad =\cfrac { \sqrt { 6 } -\sqrt { 2 } }{ 4 }$

7. Simplify $\cos { (\theta +60) } +\sin { (\theta +60) }$ $\cos { (\theta +60) } +\sin { (\theta +60) } \\ \qquad =\cos { \theta } \cos { 60 } -\sin { \theta } \sin { 60 } +\sin { \theta } \cos { 60 } +\cos { \theta } \sin { 60 } \\ \\ \qquad =\cos { \theta } \times \cfrac { 1 }{ 2 } -\sin { \theta } \times \cfrac { \sqrt { 3 } }{ 2 } +\sin { \theta } \times \cfrac { 1 }{ 2 } +\cos { \theta } \times \cfrac { \sqrt { 3 } }{ 2 } \\ \\ \qquad =\cos { \theta } \left( \cfrac { 1 }{ 2 } +\cfrac { \sqrt { 3 } }{ 2 } \right) +\sin { \theta } \left( -\cfrac { \sqrt { 3 } }{ 2 } +\cfrac { 1 }{ 2 } \right) \\ \\ \qquad =\cos { \theta } \left( \cfrac { 1+\sqrt { 3 } }{ 2 } \right) +\sin { \theta } \left( \cfrac { 1-\sqrt { 3 } }{ 2 } \right)$

8. Simplify $\cos ^{ 2 }{ 2\theta } -\sin ^{ 2 }{ 2\theta }$. $\cos ^{ 2 }{ 2\theta } -\sin ^{ 2 }{ 2\theta } \\ =\quad \cos { 2(2\theta ) } \\ =\quad \cos { 4\theta }$ 