Question 881250
<pre>
{{{matrix(1,3,  sec(a+"15°"), ""="", csc(2a)  )}}}

{{{matrix(1,3, 1/cos(a+"15°"), ""="", 1/sin(2a)  )}}}

Cross-multiply

{{{matrix(1,3,  sin(2a), ""="", cos(a+"15°")  )}}}

{{{matrix(1,7,  sin(2a), ""-"", cos(a+"15°"), "",""="","",0 )}}}

We need to get that difference to a product.  One well-known
difference to product formula is

{{{matrix(1,7,
cos(x),""-"",cos(y),"",""="","",-2sin((x+y)/2)sin((x-y)/2) )}}}

We can use it if we can write sin(2a) as a cosine.  We can
do that by using the cofunction identity, and write sin(2a) = cos(90°-2a)

{{{matrix(1,7,  sin(2a), ""-"", cos(a+"15°"), "",""="","",0 )}}}

{{{matrix(1,7,  cos("90°"-2a), ""-"", cos(a+"15°"), "",""="","",0 )}}}

Use the identity with x = 90°-2a and y = a+15°

{{{matrix(1,7,cos("90°"-2a),""-"",cos((a+"15°")),"",""="","",-2sin((("90°"-2a)+(a+"15°"))/2)sin((("90°"-2a)-(a+"15°"))/2) )}}} {{{""=""}}} {{{"0"}}}


{{{-2sin(("90°"-2a+a+"15°")/2)sin(("90°"-2a-a-"15°")/2) )}}} {{{""=""}}} {{{"0"}}}

{{{-2sin(("105°"-a)/2)sin(("75°"-3a)/2) )}}} {{{""=""}}} {{{"0"}}}

Divide both sides by -2

{{{sin(("105°"-a)/2)sin(("75°"-3a)/2) )}}} {{{""=""}}} {{{"0"}}}

Use the zero-factor property:

{{{sin(("105°"-a)/2) )}}} {{{""=""}}} {{{"0"}}}    and      {{{sin(("75°"-3a)/2) )}}} {{{""=""}}} {{{"0"}}}


{{{("105°"-a)/2) )}}} {{{""=""}}} {{{"180°"n}}}    and      {{{("75°"-3a)/2}}}{{{""=""}}} {{{"180°"n}}}

Multiply both equations through by 2

105°-a = 360°n             and       75²-3a = 360°n
      -a = 360°n-105°                   -3a = 360°n-75°
       a = -360°n+105°                    a = -120°n+25°

Both have their smallest positive value when n=0

       a = -360°(0)+105°                  a = -120°(0)+25°
       a = 105°                           a = 25°

So the smallest positive value of a is 25°

Edwin</pre>