4) Express as a single trigonometric function:
sin x(tan x + cot x)
The answerer Guest is correct but let me make it clearer.
\(sin(x)(tan(x)+cot(x))\) "Express as a single trigonometric function"
\(tan(x)=\frac{sin(x)}{cos(x)}\) (1)
\(cot(x)=\frac{cos(x)}{sin(x)}\) (2)
Substitute (1) and (2) into the equation
\(sin(x)(\frac{sin(x)}{cos(x)}+\frac{cos(x)}{sin(x)})\)
Multiply
\((\frac{sin^2(x)}{cos(x)}+cos(x))\). Now write it as a single fraction
\((\frac{sin^2(x)+cos^2(x)}{cos(x)})\)
Pythagorean identity, \(sin^2(x)+cos^2(x)=1\) substitute
\(\frac{1}{cos(x)}\)
Notice that \(sec(x)=\frac{1}{cos(x)}\)
So \(sin(x)(tan(x)+cot(x))\)\(= sec(x)\)
