If $$tan$$⁡π−$$2$$α$$4$$⋅$$tan$$⁡π−$$2$$β$$4$$⋅$$tan$$⁡π−$$2$$γ$$4=1$$, then the value of $$212($$$$sin$$⁡α⋅$$sin$$⁡β⋅$$sin$$⁡γ$$+$$$$sin$$⁡α$$+$$$$sin$$⁡β$$+$$$$sin$$⁡γ$$)$$ is

Question

If $$tan$$⁡π−$$2$$α$$4$$⋅$$tan$$⁡π−$$2$$β$$4$$⋅$$tan$$⁡π−$$2$$γ$$4=1$$, then the value of $$212($$$$sin$$⁡α⋅$$sin$$⁡β⋅$$sin$$⁡γ$$+$$$$sin$$⁡α$$+$$$$sin$$⁡β$$+$$$$sin$$⁡γ$$)$$ is

Infinity Learn · Accepted Answer

Given that $$tan$$⁡π−$$2$$α$$4tan$$⁡π−$$2$$β$$4tan$$⁡π−$$2$$γ$$4=1$$⇒$$tan2$$⁡π$$4$$−α$$2tan2$$⁡π$$4$$−β$$2tan2$$⁡π$$4$$−γ$$2=1$$⇒$$1$$−$$tan$$⁡α$$21+$$$$tan$$⁡α$$221$$−$$tan$$⁡β$$21+$$$$tan$$⁡β$$221$$−$$tan$$⁡γ$$21+$$$$tan$$⁡γ$$22=1$$⇒$$1$$−$$sin$$⁡α$$1+$$$$sin$$⁡α⋅$$1$$−$$sin$$⁡β$$1+$$$$sin$$⁡β$$1$$−$$sin$$⁡γ$$1+$$$$sin$$⁡γ$$=1$$⇒$$sin$$⁡α$$+$$$$sin$$⁡β$$+$$$$sin$$⁡γ$$+$$$$sin$$⁡α⋅$$sin$$⁡β⋅$$sin$$⁡γ$$=0$$∴$$212($$$$sin$$⁡α$$sin$$⁡β$$sin$$⁡γ$$+$$$$sin$$⁡α$$+$$$$sin$$⁡β$$+$$$$sin$$⁡γ$$)=0$$

Multiple and sub- multiple Angles

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$If \tan (\frac{π - 2 α}{4}) \cdot \tan (\frac{π - 2 β}{4}) \cdot \tan (\frac{π - 2 γ}{4}) = 1, then the value of$ $212 (\sin α \cdot \sin β \cdot \sin γ + \sin α + \sin β + \sin γ) is$

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Multiple and sub- multiple Angles

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If tan⁡π−2α4⋅tan⁡π−2β4⋅tan⁡π−2γ4=1, then the value of 212(sin⁡α⋅sin⁡β⋅sin⁡γ+sin⁡α+sin⁡β+sin⁡γ) is

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$If \tan (\frac{π - 2 α}{4}) \cdot \tan (\frac{π - 2 β}{4}) \cdot \tan (\frac{π - 2 γ}{4}) = 1, then the value of$ $212 (\sin α \cdot \sin β \cdot \sin γ + \sin α + \sin β + \sin γ) is$