2. solve cos 2x-3sin x cos 2x=0 for the principal values to two decimal places.

3. solve tan^2 + tan x-1= 0 for the principal values to two decimal places.

4. Prove that tan^2(x) -1 + cos^2(x) = tan^2(x) sin^2 (x).

5.Prove that tan(x) sin(x) + cos(x)= sec(x)

6.Prove that tan(x) cos^2(x)+sin^2(x)= cos(x)+ sin(x)

7.Prove that 1+tan(x)/1-tan(x)= sec^2(x)+ 2tan(x)/1-tan^2(x)

8.Prove that sin^2(x)-cos^2(x)/tan(x)sin(x)+cos(x)tan(x)=cos(x)-con(x)cos(x)

9. find a counterexample to show that the equation sec(x)-cos(x)=sin(x) sec(x) is not an identity

#2

cos2x - 3sinx cos2x = 0
(cos2x)(1-3sinx) = 0
as you know, if the product of two numbers is zero, one or the other must be zero. So, cos2x = 0 or 1-3sinx = 0

cos2x=0 means x is pi/4,3pi/4,5pi/4,7pi/4

1-3sinx=0 means x = arcsin(1/3) = .3398
But, you need all angles between 0 and 2pi, so since sinx >0 in Qi and QII,
x = .3398 or pi-.3398=2.8018
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#3.
did this one already also. What was unclear?
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#4
possibly the most useful trig identity is sin^2 x + cos^2 x = 1. You have

tan^2 x - 1 + cos^2 x
tan^2 x - (1-cos^2 x)
tan^2 x - sin^2 x
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#5.
tanx sinx + cosx
sinx/cosx * sinx + cosx
(sin^2 x + cos^2 x)/cosx
1/cosx
secx
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#6.
Must be a typo. If x=pi/4,
tan(x) cos^2(x)+sin^2(x) = 1*1/2 + 1/2 = 1
cos(x)+sin(x) = 1/√2+1/√2 = √2
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#7.
(1+tanx)/(1-tanx)
(1+tanx)^2 / (1-tan^2 x)
(1 + 2tanx + tan^2 x)/(1-tan^2 x)
(sec^2 x + 2tanx)/(1-tan^2 x)
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#8.
(sin^2 x - cos^2 x)/(tanx*sinx + cosx*tanx)
(sinx-cosx)(sinx+cosx)/(tanx(sinx+cosx))
(sinx-cosx)/tanx
sinx*cotx - cosx*cotx
cosx - cosx*cotx
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#9.
Usually a familiar angle will do the trick. If x=pi/4,
secx-cosx = √2 - 1/√2
sinx*secx = 1/√2*√2 = 1

2. To solve the equation cos 2x - 3sin x cos 2x = 0, we can use trigonometric identities and algebraic manipulation.

Step 1: Rewrite cos 2x using the double angle formula: cos 2x = 1 - 2sin^2 x.
The equation becomes (1 - 2sin^2 x) - 3sin x(1 - 2sin^2 x) = 0.

Step 2: Distribute and simplify: 1 - 2sin^2 x - 3sin x + 6sin^3 x = 0.
Rearranging terms, we get 6sin^3 x - 2sin^2 x - 3sin x + 1 = 0.

Step 3: Factor the equation (if possible) or use numerical methods to find the solutions. In this case, factoring is not possible, so we need to use numerical methods such as graphing or approximation methods like Newton's method or the bisection method to find the solutions.