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Subjective
Determine the smallest positive value of number $$x$$ (in degrees) for which $$\tan \left( {x + {{100}^ \circ }} \right) = \tan \left( {x + {{50}^ \circ }} \right)\,\tan \left( x \right)\tan \left( {x - {{50}^ \circ }} \right).$$$## Answer $${30^ \circ }$$ 2 ### IIT-JEE 1992 Subjective Show that the value of $${{\tan x} \over {\tan 3x}},$$ wherever defined never lies between $${1 \over 3}$$ and 3. ## Answer Solve it. 3 ### IIT-JEE 1991 Subjective If $$\exp \,\,\,\left\{ {\left( {\left( {{{\sin }^2}x + {{\sin }^4}x + {{\sin }^6}x + \,\,\,..............\infty } \right)\,In\,\,2} \right)} \right\}$$ satiesfies the equation $${x^2} - 9x + 8 = 0,$$ find the value of $${{\cos x} \over {\cos x + \sin x}},\,0 < x < {\pi \over 2}.$$ ## Answer $${{\sqrt 3 - 1} \over 2}$$ 4 ### IIT-JEE 1990 Subjective $$ABC$$ is a triangle such that $$\sin \left( {2A + B} \right) = \sin \left( {C - A} \right) = \, - \sin \left( {B + 2C} \right) = {1 \over 2}.$$$

If $$A,\,B$$ and $$C$$ are in arithmetic progression, determine the values of $$A,\,B$$ and $$C$$.

$${45^ \circ },\,\,\,{60^ \circ },\,\,\,{75^ \circ }$$

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