Introduction to...

The value of $$\displaystyle \cot 15^{\circ}\cot 16^{\circ}\cot 17^{\circ}.....\cot 73^{\circ}\cot 74^{\circ}\cot 75^{\circ}$$ is

Evaluate : $$\displaystyle \frac{3\cos 53^{\circ} \text{cosec}37^{\circ}}{(\cos ^{2}29^{\circ}+\cos ^{2}61^{\circ})}-3\tan ^{2}45^{\circ}$$

Evaluate: $$\displaystyle \tan 7^{\circ}\tan 23^{\circ}\tan 60^{\circ}\tan 67{^{\circ}}\tan 83^{\circ}$$

Let S = $$\displaystyle \sin ^{2}30^{\circ}+\sin ^{2}45^{\circ}+\sin ^{2}60^{\circ}$$ and 

Evaluate: $$\cfrac { \sin { \theta  } \cos { \theta  } \sin { \left( { 90 }^{ o }-\theta  \right)  }  }{ \cos { \left( { 90 }^{ o }-\theta  \right)  }  } +\cfrac { \cos { \theta  } \sin { \theta  } \cos { \left( { 90 }^{ o }-\theta  \right)  }  }{ \sin { \left( { 90 }^{ o }-\theta  \right)  }  } +\cfrac { \sin ^{ 2 }{ { 27 }^{ o } } +\sin ^{ 2 }{ { 63 }^{ o } }  }{ \cos ^{ 2 }{ { 40 }^{ o } } +\cos ^{ 2 }{ { 50 }^{ o } }  } $$

If $$\cos (\alpha+\beta)=0$$, then $$\sin (\alpha-\beta)$$, can be reduced to

Evaluate: $$\sin { \left( { 50 }^{ o }+\theta  \right)  } -\cos { \left( { 40 }^{ o }-\theta  \right)  } +\tan {1}^{o} \tan {10}^{o} \tan {20}^{o} \tan {70}^{o} \tan {80}^{o} \tan {89}^{o}$$

The value of $$\displaystyle \cos ^{4}\theta +\sin ^{4}\theta +2\cos ^{2}\theta \sin ^{2}\theta $$ when $$\displaystyle \theta=60^{\circ} $$ is

Evaluate: $$\cfrac { \cos ^{ 2 }{ { 20 }^{ o } } +\cos ^{ 2 }{ { 70 }^{ o } }  }{ \sec ^{ 2 }{ { 50 }^{ o } } -\cot ^{ 2 }{ { 40 }^{ o } }  } +2\text{cosec} ^{ 2 }{ { 58 }^{ o } } -2\cot { { 58 }^{ o } } \tan { { 32 }^{ o } } -4\tan { { 13 }^{ o } } \tan { { 37 }^{ o } } \tan { { 45 }^{ o } } \tan { { 53 }^{ o } } \tan { { 77 }^{ o } } $$

Evaluate:
$$\cfrac { \sec ^{ 2 }{ { 54 }^{ o } } -\cot ^{ 2 }{ { 36 }^{ o } }  }{ co\sec ^{ 2 }{ { 57 }^{ o } } -\tan ^{ 2 }{ { 33 }^{ o } }  } +2\sin ^{ 2 }{ { 38 }^{ o } } \sec ^{ 2 }{ { 52 }^{ o } } -\sin ^{ 2 }{ { 45 }^{ o } } +\cfrac { 2 }{ \sqrt { 3 }  } \tan { { 17 }^{ o } } \tan { { 60 }^{ o } } \tan { { 73 }^{ o } } $$