Extreme values of trigonometric functions in trigonometry
Question

# If  where a is constant and  $\mathrm{\alpha },\mathrm{\beta },\mathrm{\gamma }$ are variable angles. Then the least value of 2727 $\left({\mathrm{tan}}^{2}\mathrm{\alpha }+{\mathrm{tan}}^{2}\mathrm{\beta }+{\mathrm{tan}}^{2}\mathrm{\gamma }\right)$ must be

Difficult
Solution

## we have , ${\left(\mathrm{atan}\mathrm{\beta }-\sqrt{\left({\mathrm{a}}^{2}-1\right)}\mathrm{tan}\mathrm{\alpha }\right)}^{2}+\left(\sqrt{\left({\mathrm{a}}^{2}-1\right)}\mathrm{tan}\mathrm{\gamma }{-\sqrt{\left({\mathrm{a}}^{2}+1\right)}\mathrm{tan}\mathrm{\beta }\right)}^{2}+{\left(\sqrt{\left({\mathrm{a}}^{2}+1\right)}\mathrm{tan}\mathrm{\alpha }-\mathrm{atan}\mathrm{\gamma }\right)}^{2}\ge 0$$⇒\left({\mathrm{a}}^{2}+{\mathrm{a}}^{2}-1+{\mathrm{a}}^{2}+1\right)\left({\mathrm{tan}}^{2}\mathrm{\alpha }+{\mathrm{tan}}^{2}\mathrm{\beta }+{\mathrm{tan}}^{2}\mathrm{\gamma }\right)-{\left\{\mathrm{atan}\mathrm{\alpha }+\sqrt{\left({\mathrm{a}}^{2}-1\right)}\mathrm{tan}\mathrm{\beta }+\sqrt{\left({\mathrm{a}}^{2}+1\right)}\mathrm{tan}\mathrm{\gamma }\right\}}^{2}\ge 0$   [using Lagrange's identity]Hence $2727\left({\mathrm{tan}}^{2}\mathrm{\alpha }+{\mathrm{tan}}^{2}\mathrm{\beta }+{\mathrm{tan}}^{2}\mathrm{\gamma }\right)\ge 3636$Least value is 3636.

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