3 Prove that $$ \sin ^{2} x+\cos ^{2} x=1 $$ 4 Determine the general solution of $$ 2 \cos ^{2} x+5 \sin x=4 $$

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To prove that $$ \sin ^{2} x + \cos ^{2} x = 1 $$, we can use the Pythagorean identity for trigonometric functions.

The Pythagorean identity states that for any angle $$ x $$, the sum of the squares of the sine and cosine of the angle is equal to 1. This can be expressed as:

$$ \sin ^{2} x + \cos ^{2} x = 1 $$

This identity is a fundamental property of trigonometric functions and can be used to simplify expressions involving sine and cosine.

To determine the general solution of the equation $$ 2 \cos ^{2} x + 5 \sin x = 4 $$, we can use trigonometric identities and solve the equation step by step.

Let's start by solving the equation $$ 2 \cos ^{2} x + 5 \sin x = 4 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$2\cos^{2}\left(x\right)+5\sin\left(x\right)=4$$
- step1: Rewrite the expression:
  $$2-2\sin^{2}\left(x\right)+5\sin\left(x\right)=4$$
- step2: Move the expression to the left side:
  $$-2-2\sin^{2}\left(x\right)+5\sin\left(x\right)=0$$
- step3: Factor the expression:
  $$\left(-2+\sin\left(x\right)\right)\left(1-2\sin\left(x\right)\right)=0$$
- step4: Separate into possible cases:
  $$\begin{align}&-2+\sin\left(x\right)=0\&1-2\sin\left(x\right)=0\end{align}$$
- step5: Solve the equation:
  $$\begin{align}&\sin\left(x\right)=2\&\sin\left(x\right)=\frac{1}{2}\end{align}$$
- step6: Rearrange the terms:
  $$\begin{align}&x \notin \mathbb{R}\&\sin\left(x\right)=\frac{1}{2}\end{align}$$
- step7: Calculate:
  $$\begin{align}&x \notin \mathbb{R}\&x=\left\{ \begin{array}{l}\frac{\pi }{6}+2k\pi \\frac{5\pi }{6}+2k\pi \end{array}\right.,k \in \mathbb{Z}\end{align}$$
- step8: Find the union:
  $$x=\left\{ \begin{array}{l}\frac{\pi }{6}+2k\pi \\frac{5\pi }{6}+2k\pi \end{array}\right.,k \in \mathbb{Z}$$
The general solution of the equation $$ 2 \cos ^{2} x + 5 \sin x = 4 $$ is given by:

$$ x = \frac{\pi}{6} + 2k\pi \text{ or } x = \frac{5\pi}{6} + 2k\pi $$

where $$ k $$ is an integer.
To prove that $$ \sin ^{2} x + \cos ^{2} x = 1 $$, use the Pythagorean identity. For the equation $$ 2 \cos ^{2} x + 5 \sin x = 4 $$, the general solution is $$ x = \frac{\pi}{6} + 2k\pi $$ or $$ x = \frac{5\pi}{6} + 2k\pi $$, where $$ k $$ is any integer.

3 Prove that
4 Determine the general solution of

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