feminizmiki

2022-01-16

Evalute $\frac{\mathrm{tan}\alpha -\mathrm{cot}\alpha }{{\mathrm{sin}}^{4}\alpha -{\mathrm{cos}}^{4}\alpha }$ if $\alpha$ is an acute angle and $\mathrm{tan}\alpha =2$

RizerMix

Expert

Hints: $\mathrm{tan}a=\frac{1}{\mathrm{cot}a}$

Vasquez

Expert

With openspaces

user_27qwe

Expert

Factorizing the denominator and using the fact that ${\mathrm{sin}}^{2}a+{\mathrm{cos}}^{2}a=1$, we have $\frac{\mathrm{tan}\alpha -\mathrm{cot}\alpha }{{\mathrm{sin}}^{4}\alpha -{\mathrm{cos}}^{4}\alpha }=\frac{\mathrm{tan}\alpha -\mathrm{cot}\alpha }{{\mathrm{sin}}^{2}\alpha -{\mathrm{cos}}^{2}\alpha }={\mathrm{sec}}^{2}\alpha \frac{\mathrm{tan}\alpha -\mathrm{cot}\alpha }{{\mathrm{tan}}^{2}\alpha -1}=\left({\mathrm{tan}}^{2}\alpha +1\right)\frac{\mathrm{tan}\alpha -\mathrm{tan}\alpha }{{\mathrm{tan}}^{2}\alpha -1}$ $=\left(4+1\right)\frac{\left(2-\frac{1}{2}\right)}{4-1}=\frac{5}{2}$