# Solve the trigonometrical product

solve this equation :

$(2-\sec^21)(2-\sec^22)(2-\sec^23)........(2-\sec^288)(2-\sec^289)$

If tried from tangent approach with$(1+1-sec^21)......(1+1-\sec^289)$

and i do (1,89) ; (2,88);and........ so on so i get

$(1-\tan^21)(1-\tan^289)$

$(1+\tan^21\tan^289-\tan^289-\tan^21)$

i've got $\tan^21\tan^289= 1$

from $\tan(89+1)=\frac{\tan89+\tan1}{1-\tan89\tan1}$

but i don't know how to get $(\tan^289+\tan^21)$

and i don't know how to continue it

and please from other approach too

thank you

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You get $$\prod_{0\le x\le 89}(1-\tan^2x^\circ)\text{ not }\prod_{0\le x\le 89}(1+\tan^2x^\circ)$$

But $\tan45^\circ=1$

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Yes, so the answer is 0 i'm very stupid Thank you very much it's so much helping – freeze Apr 17 '13 at 8:27

Hint: What is $\cos 45^{\circ}$?

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$\frac{2^{\frac{1}{2}}}{2}$ but what is the relationship? – freeze Apr 17 '13 at 8:18
@freeze: I suggest you don't just stop there, and see how you might use that hint. Here is another puzzle of the same kind: what is $(x-a)(x-b)(x-c)\dots(x-z)?$ – Aryabhata Apr 17 '13 at 8:21
but i don't know how to get $(\tan^289+tan^21)$? can you help me – freeze Apr 17 '13 at 8:24
@freeze: What's the value of this : $(4-1)(4-2)(4-3)(4-4)(4-5)..(4-n)$? – Inceptio Apr 17 '13 at 9:34