Question 1192560
.
Determine the constants a and b so that {{{ (-3+4 cos^(2) x)/ (1-2 sin x) = a+b sin x }}} for all values of x.
~~~~~~~~~~~~~~~~~


<pre>
Transform the numerator step by step

    {{{-3 + 4*cos^2(x)}}} = {{{-3 + 4*(1-sin^2(x))}}} = {{{-3 + 4 - 4*sin^2(x)}}} = {{{1 - 4*sin^2(x)}}} = {{{(1-2sin(x))*(1+2sin(x))}}}.


So, your fraction now is

    {{{(-3+4 cos^2(x))/(1-2*sin(x))}}} = {{{((1-2sin(x))*(1+2sin(x)))/(1-2*sin(x))}}}.


Cancel the factors ((1-2*sin(x)) in both the numerator and denominator.  You will get then

    {{{(-3+4 cos^2(x))/(1-2*sin(x))}}} = {{{((1-2sin(x))*(1+2sin(x)))/(1-2*sin(x))}}} = 1 + 2*sin(x).


It is just the form which you need.  So,  a= 1,  b= 2.


<U>ANSWER</U>.  a= 1,  b= 2.
</pre>

Solved.