Question

Prove sec x - tan x sin x = (1/sec x) ?

Answer 1

See my proof below

Explanation:

We will simplify the left-hand side of your equation:
`sec(x)-tan(x)*sec(x)=`

`1/cos(x)-sin^2(x)/cos(x)=(1-sin^2(x))/cos(x)`

(since `tan(x)*sin(x)=sin(x)/cos(x)*sin(x)=sin^2(x)/cos(x)`)

further

`(1-sin^2(x))/cos(x)=cos^2(x)/cos(x)=cos(x)=1/sec(x)`

(since `1-sin^2(x)=cos^2(x)`)

Answer 2

`"see explanation"`

Explanation:

`"using the "color(blue)"trigonometric identities"`

`•color(white)(x)cosx=1/secxhArrsecx=1/cosx`

`•color(white)(x)tanx=sinx/cosx`

`•color(white)(x)cos^2x+sin^2x=1`

`"consider the left side"`

`1/cosx-sinx/cosx xxsinx`

`=1/cosx-sin^2x/cosx`

`=(1-sin^2x)/cosx`

`=cos^2x/cosx`

`=cosx=1/secx=" right side "rArr" verified"`