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  }

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Examples:

1. Simplify \sin { \theta  } cot\theta .

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\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.

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\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  }  }

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\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 } .

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\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 }  }.

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\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 } .

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\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) }

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\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 } .

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\cos ^{ 2 }{ 2\theta } -\sin ^{ 2 }{ 2\theta } \\ =\quad \cos { 2(2\theta ) } \\ =\quad \cos { 4\theta }

 

And more examples of Double Angle and Half Angle problems:

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