Cho 3 sin^4 x - cos^4 x = 1/2. Tính A = 2sin^4x - cos^4x
Câu hỏi:
Cho 3sin4x – cos4x = \(\frac{1}{2}.\) Tính A = 2sin4x – cos4x.
Trả lời:
Ta có: sin2x + cos2x = 1
⇔ cos2x = 1 – sin2x
Mà 3sin4x – cos4x = \(\frac{1}{2}.\)
⇔ 3sin4x – (1 – sin2x)2 = \(\frac{1}{2}.\)
⇔ 3sin4x – 1 + 2sin2x – sin4x – \(\frac{1}{2}\)= 0
⇔ 2sin4x + 2sin2x – \(\frac{3}{2}\) = 0
⇔ \(\left[ \begin{array}{l}{\sin ^2}x = \frac{1}{2}\\{\sin ^2}x = \frac{{ - 3}}{2}\left( L \right)\end{array} \right.\)
⇔ sin2x = \(\frac{1}{2}\)
⇔ cos2x = \(\frac{1}{2}\)
A = 2sin4x – cos4x = \(2.{\left( {\frac{1}{2}} \right)^2} - {\left( {\frac{1}{2}} \right)^2} = \frac{1}{4}\).
