Sum and Difference Trigonometric Formulas

Sum and Difference Trigonometric Formulas: Level 4 Challenges


\[\large \sin^{6}(\theta)+\cos^{6}(\theta)=\sin(2\theta) = \ ? \]

If the above equation is true, then what is the value of \(\sin(2\theta)\)? Give your answer to 3 decimal places.

\[\Large\log_{\frac12}{\left(\displaystyle\prod_{n=1}^{45}\sin{(2n-1)}^{\circ}\right)}=\ ?\]

\[ \tan \left [ \tan^{-1}\left(\frac12\right)+\tan^{-1}\left(\frac29\right)+\tan^{-1}\left(\frac18\right)+\tan^{-1}\left(\frac2{25}\right)+\tan^{-1}\left(\frac1{18}\right)+\ldots \right] \]

What is the value of the expression above?

\[ \left(\sec^{2}10^{\circ}+\tan 10^{\circ}\right)\left(\sec^{2} 50^{\circ} - \tan 50^{\circ}\right)\left(\sec^{2} 70^{\circ}+\tan 70^{\circ}\right)\]

The value of the expression above can be represented in the form \(\dfrac{a-\sqrt{b}}{b}\) where \(a\) and \(b\) are positive coprime integers. Find the value of \(a+b^2\).

\[\large \displaystyle\prod_{r=1}^{44} (\cot (r^\circ) - 1)\]

This product equals \(2^{k}.\) Find \(k.\)


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