How to write in terms of sin ? ; tan(x) + cot (x) / sec (x)

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Answer 1 · 2018-06-12T11:40:15

Please see the explanation below

We need

$tan x = \frac{sin x}{cos x}$ $tanx=sinx/cosx$

$cot x = \frac{cos x}{sin x}$ $cotx=cosx/sinx$

$sec x = \frac{1}{cos x}$ $secx=1/cosx$

${sin}^{2} x + {cos}^{2} x = 1$ $sin^2x+cos^2x=1$

The expression is

$\frac{tan x + cot x}{sec x} = \frac{\frac{sin x}{cos x} + \frac{cos x}{sin x}}{\frac{1}{cos x}}$ $(tanx+cotx)/secx=(sinx/cosx+cosx/sinx)/(1/cosx)$

$= \frac{{sin}^{2} x + {cos}^{2} x}{sin x cos x} \cdot cos x$ $=(sin^2x+cos^2x)/(sinxcosx)*cosx$

$= \frac{1}{sin x cos x} \cdot cos x$ $=1/(sinxcosx)*cosx$

$= \frac{1}{sin x}$ $=1/sinx$