Answer
(a)
$\frac{d}{dx}tanx$ = $\frac{d}{dx}\frac{sinx}{cosx}$
$sec^{2}x$ = $\frac{cosxcosx+sinxsinx}{cos^{2}x}$
$sec^{2}x$ = $\frac{cos^{2}x+sin^{2}x}{cos^{2}x}$
$sec^{2}x$ = $\frac{1}{cos^{2}x}$
$sec^{2}x$ = $sec^{2}x$
(b)
$\frac{d}{dx}secx$ = $\frac{d}{dx}\frac{1}{cosx}$
$secxtanx$ = $\frac{cosx(0)+(1)(-sinx)}{cos^{2}x}$
$secxtanx$ = $\frac{sinx}{cos^{2}x}$
$secxtanx$ = $secxtanx$
(c)
$\frac{d}{dx}(sinx+cosx)$ = $\frac{d}{dx}\frac{1+cotx}{cscx}$
$\frac{d}{dx}(cosx-sinx)$ = $\frac{cscx(-csc^{2}x)-(1+cotx)(-cscxcotx)}{csc^{2}x}$
$\frac{d}{dx}(cosx-sinx)$ = $\frac{-csc^{2}x+-cot^{2}x+cotx}{cscx}$
$\frac{d}{dx}(cosx-sinx)$ = $\frac{-1+cotx}{cscx}$
$\frac{d}{dx}(cosx-sinx)$ = $\frac{cotx-1}{cscx}$
$\frac{d}{dx}(cosx-sinx)$ = $\frac{d}{dx}(cosx-sinx)$
Work Step by Step
(a)
$\frac{d}{dx}tanx$ = $\frac{d}{dx}\frac{sinx}{cosx}$
$sec^{2}x$ = $\frac{cosxcosx+sinxsinx}{cos^{2}x}$
$sec^{2}x$ = $\frac{cos^{2}x+sin^{2}x}{cos^{2}x}$
$sec^{2}x$ = $\frac{1}{cos^{2}x}$
$sec^{2}x$ = $sec^{2}x$
(b)
$\frac{d}{dx}secx$ = $\frac{d}{dx}\frac{1}{cosx}$
$secxtanx$ = $\frac{cosx(0)+(1)(-sinx)}{cos^{2}x}$
$secxtanx$ = $\frac{sinx}{cos^{2}x}$
$secxtanx$ = $secxtanx$
(c)
$\frac{d}{dx}(sinx+cosx)$ = $\frac{d}{dx}\frac{1+cotx}{cscx}$
$\frac{d}{dx}(cosx-sinx)$ = $\frac{cscx(-csc^{2}x)-(1+cotx)(-cscxcotx)}{csc^{2}x}$
$\frac{d}{dx}(cosx-sinx)$ = $\frac{-csc^{2}x+-cot^{2}x+cotx}{cscx}$
$\frac{d}{dx}(cosx-sinx)$ = $\frac{-1+cotx}{cscx}$
$\frac{d}{dx}(cosx-sinx)$ = $\frac{cotx-1}{cscx}$
$\frac{d}{dx}(cosx-sinx)$ = $\frac{d}{dx}(cosx-sinx)$
