Question

How do you prove: `secx - cosx = sinx tanx`?

Answer 1

Using the definitions of `secx` and `tanx`, along with the identity
`sin^2x + cos^2x = 1`, we have

`secx-cosx = 1/cosx-cosx`

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

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

`=sin^2x/cosx`

`=sinx *sinx/cosx`

`=sinxtanx`

Answer 2

First convert all terms into `sinx` and `cosx`.
Second apply fraction sum rules to the LHS.
Lastly we apply the Pythagorean identity: `sin^2 x + cos^2 x =1`

Explanation:

First in questions of these forms it's a good idea to convert all terms into sine and cosine: so, replace `tan x` with `sin x /cos x`
and replace `sec x` with `1/ cos x`.

The LHS, `sec x- cos x` becomes `1/cos x- cos x`.
The RHS, `sin x tan x` becomes `sin x sin x/cos x` or `sin^2 x / cos x`.

Now we apply fraction sum rules to the LHS, making a common base (just like number fraction like `1/3 +1/4 => 4/12 + 3/12 = 7/12)`.
LHS=`1/cos x- cos x => 1/cos x- cos^2 x/cos x => {1 - cos^2 x} /cos x`.

Lastly we apply the Pythagorean identity: `sin^2 x + cos^2 x =1`! (one of the most useful identities for these types of problems).
By rearranging it we get `1- cos^2 x = sin^2 x`.
We replace the `1- cos^2 x` in the LHS with `sin^2 x`.

LHS = `{1 - cos^2 x} /cos x => {sin^2 x} /cos x` which is equal to the modified RHS.

Thus LHS= RHS Q.E.D.

Note this general pattern of getting things into terms of sine and cosine, using the fraction rules and the Pythagorean identity, often solves these types of questions.

Answer 3

If we so desire, we can also modify the right-hand side to match the left-hand side.

We should write `sinxtanx` in terms of `sinx` and `cosx`, using the identity `color(red)(tanx=sinx/cosx)`:

`sinxtanx=sinx(sinx/cosx)=sin^2x/cosx`

Now, we use the Pythagorean identity, which is `sin^2x+cos^2x=1`. We can modify this to solve for `sin^2x`, so: `color(red)(sin^2x=1-cos^2x)`:

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

Now, just split up the numerator:

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

Use the reciprocal identity `color(red)(secx=1/cosx`:

`1/cosx-cosx=secx-cosx`

Answer 4

It's really this simple...

Explanation:

Using the identity `tanx=sinx/cosx`, multiply the `sinx` onto the identity to get:

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

Then, multiply `cosx` through the equation to yield:

`1-cos^2x=sin^2x`

Considering that `secx` is the inverse of `cosx`.
Finally, using the trigonometric identity `1-cos^2x=sin^2x`, the final answer would be:

`sin^2x=sin^2x`