MHB How Can You Prove This Trigonometric Identity?

AI Thread Summary
The discussion focuses on proving the trigonometric identity $(4\cos^2 9^{\circ}-3)(4\cos^2 27^{\circ}-3)=\tan 9^{\circ}$. Participants highlight that familiarity with the triple angle formula for cosine is key to solving the problem efficiently. The problem is considered manageable, with encouragement for those engaging with it. Overall, the conversation emphasizes the importance of understanding trigonometric identities and formulas in solving such challenges. Mastery of these concepts is essential for proving complex trigonometric identities.
anemone
Gold Member
MHB
POTW Director
Messages
3,851
Reaction score
115
Prove that $(4\cos^2 9^{\circ}-3)(4\cos^2 27^{\circ}-3)=\tan 9^{\circ}$
 
Mathematics news on Phys.org
anemone said:
Prove that $(4\cos^2 9^{\circ}-3)(4\cos^2 27^{\circ}-3)=\tan 9^{\circ}$

Using the fact that
$$\cos(3x)=4\cos^3x-3\cos x\,\,\,\,(*)$$
LHS can be written as:
$$\frac{\cos 27^{\circ}}{\cos 9^{\circ}}(4\cos^2 27^{\circ}-3)=\frac{4\cos^3 27^{\circ}-3\cos 27^{\circ}}{\cos 9^{\circ}}$$
Again by using (*),
$$\frac{4\cos^3 27^{\circ}-3\cos 27^{\circ}}{\cos 9^{\circ}}=\frac{\cos 81^{\circ}}{\cos 9^{\circ}}=\frac{\sin 9^{\circ}}{\cos 9^{\circ}}=\tan 9^{\circ}$$
 
anemone said:
Prove that $(4\cos^2 9^{\circ}-3)(4\cos^2 27^{\circ}-3)=\tan 9^{\circ}$

we know

$\cos 3x = 4 cos ^3 x - 3 cos x$

so $\frac{\cos 3x}{\cos x} = 4 cos ^2 x - 3$

put $x = 9^{\circ}$ to get $(4\cos^2 9^{\circ}-3)= \cos 27^{\circ}/\cos 9^{\circ}$

put $x = 27^{\circ}$ to get $(4\cos^2 27^{\circ}-3)= \cos 81^{\circ}/\cos 27^{\circ}$

hence $(4\cos^2 9^{\circ}-3)(4\cos^2 27^{\circ}-3)$
=
$\cos 81^{\circ}/\cos 27^{\circ}\cos 27^{\circ}/\cos 9^{\circ}$
=$ \cos 81^{\circ}/\cos 9^{\circ}$
= $ \sin 9^{\circ}/\cos 9^{\circ}$
= $ tan 9^{\circ}$
 
Pranav said:
Using the fact that
$$\cos(3x)=4\cos^3x-3\cos x\,\,\,\,(*)$$
LHS can be written as:
$$\frac{\cos 27^{\circ}}{\cos 9^{\circ}}(4\cos^2 27^{\circ}-3)=\frac{4\cos^3 27^{\circ}-3\cos 27^{\circ}}{\cos 9^{\circ}}$$
Again by using (*),
$$\frac{4\cos^3 27^{\circ}-3\cos 27^{\circ}}{\cos 9^{\circ}}=\frac{\cos 81^{\circ}}{\cos 9^{\circ}}=\frac{\sin 9^{\circ}}{\cos 9^{\circ}}=\tan 9^{\circ}$$

kaliprasad said:
we know

$\cos 3x = 4 cos ^3 x - 3 cos x$

so $\frac{\cos 3x}{\cos x} = 4 cos ^2 x - 3$

put $x = 9^{\circ}$ to get $(4\cos^2 9^{\circ}-3)= \cos 27^{\circ}/\cos 9^{\circ}$

put $x = 27^{\circ}$ to get $(4\cos^2 27^{\circ}-3)= \cos 81^{\circ}/\cos 27^{\circ}$

hence $(4\cos^2 9^{\circ}-3)(4\cos^2 27^{\circ}-3)$
=
$\cos 81^{\circ}/\cos 27^{\circ}\cos 27^{\circ}/\cos 9^{\circ}$
=$ \cos 81^{\circ}/\cos 9^{\circ}$
= $ \sin 9^{\circ}/\cos 9^{\circ}$
= $ tan 9^{\circ}$

Hi Pranav and kaliprasad,:)

Thank you so much for participating in this not very difficult challenge trigonometric problem and well done! Yes, the trick to prove this identity quickly lies with the fact if one is familiar with the triple angle formula for cosine function.(Muscle)
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...

Similar threads

Back
Top