# Maths - Trigonometry Identities

> Source: https://ollybritton.com/notes/a-level/maths/topics/trigonometry-identities/ · Updated: 2024-08-25 · Tags: maths, further-maths

##### How could you rewrite the hypotenuse in terms of $\cos$ and $\sin$??
$$
\frac{\sin \theta}{\cos \theta}
$$

##### In a circle with width $\sin$ and height $\cos$, what could you write because of Pythagorus??
$$
\sin^2\theta + \cos^2\theta = 1
$$

##### How could you simplify $7(1 - \cos^2\theta)$??
$$
7\sin^2\theta
$$

##### How could you rewrite $(\cos^4\theta - \sin^4\theta)$ as the difference of two squares??
$$
(\cos^2\theta - \sin^2\theta)(\cos^2\theta + \sin^2\theta)
$$

##### How could you simplify $(\cos^2\theta - \sin^2\theta)(\cos^2\theta + \sin^2\theta)$??
$$
(\cos^2\theta - \sin^2\theta)
$$

##### How could you rewrite $3\sin\theta - 2\cos\theta = 0$??
$$
\tan\theta = \frac{2}{3}
$$

##### What is the formula for $\sin(A \pm B)$??
$$
\sin A\cos B \pm \cos A\sin B
$$

##### What is the formula for $\cos(A \pm B)$??
$$
\cos A\cos B \mp \sin A\sin B
$$

##### What is the formula for $\tan(A \pm B)$??
$$
\frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}
$$

##### What is the formula for $\sin2A$??
$$
2\sin A\cos A
$$

##### What is the formula for $\cos2A$??
$$
2\cos^2 A - 1
$$

##### What is the formula for $\tan2A$??
$$
\frac{2\tan A}{1-\tan^2 A}
$$

### 2021-11-09
##### What are the three formulas for $\cos 2\theta$??
$$
\cos^2 \theta - \sin^2 \theta
$$
$$
2\cos^2 \theta - 1
$$
$$
1-2\sin^2 \theta
$$

### 2022-05-03
##### $$\sin(P) + \sin(Q) = 2\sin\left(\frac{P + Q}{2}\right)\cos\left(\frac{P - Q}{2}\right)$$ How can you go about proving this??
Start by considering
$$
\sin(A + B) + \sin(A - B)
$$

and then make a substitution.

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