# Trigonometric identities involved $\sin(\alpha)$, $\cos(\alpha)$, $\tan(\alpha)$

$\alpha$ is valid between $-90<\alpha<90$ Degrees

Show that,

$$\frac{\tan^6(\alpha)-\tan^4(\alpha)+2}{\tan^6(\alpha)-2\tan^2(\alpha)+4 }\cdot\cos^2(\alpha) = \frac{\sin^6(\alpha)+\sin^4(\alpha)-2}{\sin^6(\alpha)-2\sin^2(\alpha)-4 }$$

$$\frac{\sin^6(\alpha)+\sin^4(\alpha)-2}{\sin^6(\alpha)-2\sin^2(\alpha)-4 } =\frac{(\sin^2(\alpha)-1)(\sin^4(\alpha)+2\sin(\alpha)+2)} {(\sin^2(\alpha)-4)(\sin^4(\alpha)+2\sin(\alpha)+2)}$$

$$\frac{\sin^6(\alpha)+\sin^4(\alpha)-2}{\sin^6(\alpha)-2\sin^2(\alpha)-4 } =\frac{\sin^2(\alpha)-1} {\sin^2(\alpha)-4}$$

Can this expression be simplify more further? $$\frac{\tan^6(\alpha)-\tan^4(\alpha)+2}{\tan^6(\alpha)-2\tan^2(\alpha)+4 }\cdot\cos^2(\alpha)$$

please give a hand here can't seem to do it

• It is always a good idea to start by changing the tangent and cotangents into sine and cosines.Pythagorean identities can help too. – user242559 May 9 '16 at 22:47
• numerators and denominators can be factorized – G Cab May 9 '16 at 22:55
• How? I can't see it please show me. – user335850 May 9 '16 at 22:58
• @pisquare e.g. : $x^{\,6} + x^{\,4} - 2 = \left( {x - 1} \right)\left( {x + 1} \right)\left( {2x^{\,2} + x^{\,4} + 2} \right)$ and similarly for the other three terms, and the fractions gets much simplified. – G Cab May 9 '16 at 23:20
• @G Cab thank, $1+1-2=0$, then used division of polynomial. As for the other I can't see any values to make the equation zero! If you do show me another example, but I doubt it. – user335850 May 10 '16 at 3:55

So, for the LH fraction: $${{x^{\,6} - x^{\,4} + 2} \over {x^{\,6} - 2x^{\,2} + 4}} = {{y^{\,3} - y^{\,2} + 2} \over {y^{\,3} - 2y + 4}}$$ then by Ruffini's method $$= {{\left( {y + 1} \right)\left( {y^{\,2} - 2y^{\,2} + 1} \right)} \over {\left( {y + 2} \right)\left( {y^{\,2} - 2y^{\,2} + 1} \right)}} = {{x^{\,2} + 1} \over {x^{\,2} + 2}}$$ Same method for the RH fraction, as already done, but corrected to$${{\sin ^{\,2} \alpha - 1} \over {\sin ^{\,2} \alpha - 2}}$$