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Sum and Difference Trigonometric Formulas

Sum and Difference Trigonometric Formulas: Level 4 Challenges


sin6(θ)+cos6(θ)=sin(2θ)= ?\large \sin^{6}(\theta)+\cos^{6}(\theta)=\sin(2\theta) = \ ?

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

log12(n=145sin(2n1))= ?\Large\log_{\frac12}{\left(\displaystyle\prod_{n=1}^{45}\sin{(2n-1)}^{\circ}\right)}=\ ?

tan[tan1(12)+tan1(29)+tan1(18)+tan1(225)+tan1(118)+] \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?

(sec210+tan10)(sec250tan50)(sec270+tan70) \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 abb\dfrac{a-\sqrt{b}}{b} where aa and bb are positive coprime integers. Find the value of a+b2a+b^2.

r=144(cot(r)1)\large \displaystyle\prod_{r=1}^{44} (\cot (r^\circ) - 1)

This product equals 2k.2^{k}. Find k.k.


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