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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...
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
Thanks, now I know how to prove the identity for question 2. :D
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?
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.
oo... so the values of x=0, 180, and 360 are obtained when we take sin²(x)=0
Now my prob is solved, thanks :bigyello: