Question

express sin2x and cos2x in terms of tanx​

Answer 1

Step-by-step explanation:

Expand out sin 2x, using the double angle formula, you get:

2 sin x cos x

Now multiply through by cos x / cos x, giving:

2 sin x cos^2 x / cos x

Which can be written:

2 (sin x / cos x) * cos^2 x

We know that cos = 1 / sec, so, this can be written:

2 (sin x / cos x) * 1 / sec^2 x

We know sin x / cos x = tan x, and sec^2 = 1 + tan^2, so, we get:

sin 2x = 2 tan x / (1 + tan^2 x)

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Expand out cos 2x using the double angle formula as 2 cos^2 x - 1

We know cos = 1 / sec, so, this becomes:

(2 / sec^2 x) - 1

We know sec^2 = 1 + tan^2, so:

2 / (1 + tan^2 x) - 1

You can them simplify, as follows, by putting it all over a common denominator:

2 / (1 + tan^2 x) - (1 + tan^2 x) / (1 + tan^2 x)

= (2 - (1 + tan^2 x)) / (1 + tan^2 x)

= (1 - tan^2 x) / (1 + tan^2 x)

So we have:

→sin 2x = 2 tan x / (1 + tan^2 x)

→cos 2x = (1 - tan^2 x) / (1 + tan^2 x)