Question

(Step to step work) sin x(cotx + tanx)

Answer 1

solving $sin(x)(cot(x) + tan(x))$ we get $sec(x)$

Step-by-step explanation:

We need to solve: $sin(x)(cot(x) + tan(x))$

Solving:

We know:

$cot(x)=\frac{cos(x)}{sin(x)}\\ and\,\,tan(x)=\frac{sin(x)}{cos(x)}$

Putting the values of cot(x) and tan(x):

$sin(x)(cot(x) + tan(x))\\=sin(x)(\frac{cos(x)}{sin(x)}+\frac{sin(x)}{cos(x)})$

Taking LCM:

$=sin(x)(\frac{cos(x)cos(x)+sin(x)sin(x)}{sin(x)cos(x)})\\=sin(x)(\frac{cos^2(x)+sin^2(x)}{sin(x)cos(x)})$

We know that: cos^2x+sin^2x=1

and using identity: cos(x)sin(x)=sin(2x)*1/2

$=sin(x)(\frac{1}{sin(2x)\frac{1}{2}})\\=sin(x)(\frac{2}{sin(2x)})$

Using identity: sin(2x)=2cos(x)sin(x)

$=sin(x)(\frac{2}{2sin(x)cos(x)})\\Cancelling\,\,same\,\,terms\,\,in\,\,numerator\,\,and\,\,denominator:\\=\frac{1}{cos(x)} \\=sec(x)$

So, solving $sin(x)(cot(x) + tan(x))$ we get $sec(x)$

Keywords: Trigonometry

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