# Given sinx/cosx=2 determine cos^2x.

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We have to find (cos x)^2, given that sin x / cos x = 2

sin x / cos x = 2

square both the sides

=> (sin x)^2/ (cos x)^2 = 4

=> (1 - (cos x)^2) / (cos x)^2 = 4

=> 1/(cos x)^2 - 1 = 4

=> 1/(cos x)^2 = 5

=> (cos x)^2 = 1/5.

**Therefore (cos x)^2 = 1/5.**

sinx/cosx = 2.

To fins cos^2x.

We know sinx /cosx = tanx = 2.

We know that cosx = 1/(1+tan^2x)^(1/2)

cos^2x = 1/(1+tan^2x).

Therefore cos^x = 1/(1+2^2) = 1/(1+4) = 1/5.

Therefore cos^2x = 1/5.

We'll begin from the fundamental formula of trigonometry:

(sin x)^2 + (cos x)^2 = 1

We'll divide by (cos x)^2 both sides:

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

Since the ratio sin x/cos x = 2, we'll raise to square both sides:

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

(sin x)^2/ (cos x)^2 = 4 (2)

We'll substitute (2) in (1):

4 + 1 = 1/(cos x)^2

1/(cos x)^2 = 5

**(cos x)^2 = 1/5**