[RESOLVED] Trigonometry Problem...

[MathBaby]

I have some problem with 2 trigonometry questions..

1. Solve this equation for x, interval 0≤x≤360
4 sin²(x) cos²(x) = tan²(x)

2. Prove this identity
1 - 2 sin²(x) / cos(x) + sin(x) ≡cos(x) - sin(x)


It took me 4 hours trying to solve these 2 questions, I hope someone can teach me how to solve it...

[Glaysher]

2. Requires good knowledge of trig identities to spot that the top of the fraction is cos(2x) which can be rewritten as cos²(x) - sin²(x). This factorises and cancels with the bottom to get the right hand side

1. Multiply both sides by cos²(x) and rewrite sin²(x) in terms of cos²(x)
Then solve the resulting polynomial in cos (x) and solve the resulting trig equations

[MathBaby]

Thanks, now I know how to prove the identity for question 2.

As for the 1st question, after I multiply both sides by cos²(x) and rewrite sin²(x) in terms of cos²(x), I ended up with something like this:

4[1-cos²(x)][cos³(x)] = 1-cos²(x)

I have no idea what should I do next..

Actually I tried to divide both sides by sin²(x), then I ended up with
cos³(x) = 1/4. That gives me x= 51.0 and 309.0, but the answer for this question is x=0, 51.0, 180, 309.0 and 360

I wonder how can I get the x=0, 180, and 360 value?

[Glaysher]

Actually dividing by sin²(x) is a more elegant solution but you need to account for the possibility that sin²(x) = 0 as this would make both sides equal zero and therefore is a valid solution. This will give you the other x values.

[MathBaby]

oo... so the values of x=0, 180, and 360 are obtained when we take sin²(x)=0

Now my prob is solved, thanks