Question

(1)sin19/sin71-cos11/cos19=........................
(2) show that,√3.cos23 - sin23/2=cos53.
(3)Prove that cosФ. cosecФ = cotФ 

Answer 1

 3) Cos . 1/ sin= cot
        cot=cot
          hence proved

Answer 2

please check if the numbers given in the first problem are correct or not.

$\frac{Sin 19 }{Sin 71}-\frac{Cos11}{cos 19}\\\\=\frac{Sin 19 }{Cos(90-71)}-\frac{Cos11}{cos 19}\\\\=\frac{Sin19\ -\ Cos11}{Cos19}$

If there is Sin 79  instead of  Sin 19 in the first ratio, then the answer would be zero.  Because, Sin 79 = Cos (90 - 79) = Cos 11.   Or, perhaps there is Cos 71 instead of Cos 11 in the second ratio.  Then also the answer is 0.

 If there is no mistake in the given problem, then:

$\frac{Sin19\ -\ Cos11}{Cos19}\\\\=tan19-\frac{Sin(90-11)}{Cos19}\\\\=tan19-\frac{sin(60+19)}{cos19}\\\\tan19-\frac{sin60\ cos19+cos60\ sin19}{cos19}\\\\=tan19-\frac{\sqrt3}{2}-\frac{1}{2}tan19\\\\=-\frac{\sqrt3-tan19}{2}$
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One must use parentheses () [] {}  to clearly write the expressions,  Otherwise, we have confusion in understanding.

$\frac{\sqrt3\ cos23-sin23}{2}=\frac{\sqrt3}{2} cos23}-\frac{1}{2}Sin23\\\\=Cos30\ Cos23-Sin30 \ Sin23\\\\Cos(30+23)=Cos53$
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         CosФ * Cosec Ф
         Cos Ф * 1/ Sin Ф
        = Cos Ф / Sin Ф  = Cot Ф