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Write cos2x as sinx
please help with this ​

1 Answer

3 votes

Answer:


\cos(2\, x) = 1 - 2\, (\sin(x))^2.

Explanation:

Angle sum identity for cosine:
\cos(a + b) = \cos(a) \, \cos(b) - \sin(a) \, \sin(b).

Pythagorean identity:
(\cos(a))^(2) + (\sin(a))^(2) = 1 for all real
a.

Subtract
(\cos(x))^(2) from both sides of the Pythagorean identity to obtain:
(\sin(a))^(2) = 1 - (\cos(a))^(2).

Apply angle sum identity to rewrite
\cos(2\, x).


\begin{aligned}&\cos(2\, x)\\ &= \cos(x + x) \\ &= \cos(x) \, \cos(x) - \sin(x)\, \sin(x) \\ &= (\cos(x))^(2) + (\sin(x))^(2)\end{aligned}.


(\sin(a))^(2) = 1 - (\cos(a))^(2) follows from the Pythagorean identity. Hence, it would be possible to replace the
(\cos(x))^(2) in the previous expression with
(1 - (\sin(x))^(2)).


\begin{aligned}&(\cos(x))^(2) - (\sin(x))^(2)\\ &= \left[1 - (\sin(x))^(2)\right] - (\sin(x))^(2) \\ &= 1 - 2\, (\sin(x))^(2) \end{aligned}.

Conclusion:


\begin{aligned}&\cos(2\, x) \\ &= (\cos(x))^(2) + (\sin(x))^(2) \\ &=1 - 2\, (\sin(x))^(2)\end{aligned}

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