sin4theta - cos4theta=
Answers
sin4A - cos4A = ____
sin4A - (1 - sin2A)2
sin4A - 1 + sin2A + 2.sinA.
...........
Answer:
sin
4
θ+cos
4
θ
=1
Step-by-step explanation:
\begin{gathered}LHS= \frac{sin^{4}\theta+cos^{4}\theta}{1-2sin^{2}\theta cos^{2}\theta}\\=\frac{(sin^{2}\theta)^{2}+(cos^{2}\theta)^{2}}{1-2sin^{2}\theta cos^{2}\theta}\\=\frac{(sin^{2}\theta+cos^{2}\theta)^{2}-2sin^{2}\theta cos^{2}\theta }{1-2sin^{2}\theta cos^{2}\theta}\\\end{gathered}
LHS=
1−2sin
2
θcos
2
θ
sin
4
θ+cos
4
θ
=
1−2sin
2
θcos
2
θ
(sin
2
θ)
2
+(cos
2
θ)
2
=
1−2sin
2
θcos
2
θ
(sin
2
θ+cos
2
θ)
2
−2sin
2
θcos
2
θ
/* By algebraic identity:
i )a²+b² = (a+b)²-2ab
By Trigonometric identity:
ii) sin²A+cos²A = 1 */
\begin{gathered}= \frac{1-2sin^{2}\theta cos^{2}\theta }{1-sin^{2}\theta cos^{2}\theta }\\=1 \\=RHS\end{gathered}
=
1−sin
2
θcos
2
θ
1−2sin
2
θcos
2
θ
=1
=RHS
Therefore,
\frac{sin^{4}\theta+cos^{4}\theta}{1-2sin^{2}\theta cos^{2}\theta}=1
1−2sin
2
θcos
2
θ
sin
4
θ+cos
4
θ
=1
•••♪