x = π/4 + nπ

<p>To solve the equation sin(x) + cos(x) = 1, we can use trigonometric identities.</p><p>We know that sin^2(x) + cos^2(x) = 1 (the Pythagorean identity).</p><p>Rearranging the equation, we get:</p><p>sin(x) = 1 - cos(x)</p><p>Squaring both sides of the equation, we get:</p><p>sin^2(x) = (1 - cos(x))^2</p><p>Expanding the right side of the equation, we get:</p><p>sin^2(x) = 1 - 2cos(x) + cos^2(x)</p><p>Substituting sin^2(x) = 1 - cos^2(x) (another trigonometric identity), we get:</p><p>1 - cos^2(x) = 1 - 2cos(x) + cos^2(x)</p><p>Simplifying, we get:</p><p>2cos^2(x) - 2cos(x) = 0</p><p>Factoring out 2cos(x), we get:</p><p>2cos(x)(cos(x) - 1) = 0</p><p>So either cos(x) = 0 or cos(x) = 1.</p><p>If cos(x) = 0, then x = (2n+1)π/2, where n is an integer.</p><p>If cos(x) = 1, then x = 2nπ, where n is an integer.</p><p>Therefore, the solutions to the equation sin(x) + cos(x) = 1 are:</p><p>x = (2n+1)π/2 or x = 2nπ, where n is an integer.</p>

<p>We can solve this equation by using the trigonometric identity:</p><p>sin(x) + cos(x) = sqrt(2) * sin(x + pi/4)</p><p>Substituting this identity into the equation, we get:</p><p>sqrt(2) * sin(x + pi/4) = 1</p><p>Dividing both sides by sqrt(2), we get:</p><p>sin(x + pi/4) = 1/sqrt(2)</p><p>Using the inverse sine function, we can find the values of x that satisfy this equation:</p><p>x + pi/4 = pi/4 + 2k*pi + pi/4</p><p>or</p><p>x + pi/4 = 3<em>pi/4 + 2k</em>pi</p><p>where k is an integer.</p><p>Solving for x, we get:</p><p>x = -pi/4 + 2k<em>pi or x = 3</em>pi/4 + 2k*pi</p><p>So the solutions to the equation are:</p><p>x = -pi/4 + 2k<em>pi or x = 3</em>pi/4 + 2k*pi</p><p>where k is an integer.</p>

<p>To solve the equation sin(x) + cos(x) = 1, we can use trigonometric identities and techniques.</p><p>First, we can rewrite the equation as sin(x) + cos(x) - 1 = 0.</p><p>Next, we can use the identity sin^2(x) + cos^2(x) = 1, which implies that sin^2(x) = 1 - cos^2(x).</p><p>Substituting this into the equation, we have (1 - cos^2(x)) + cos(x) - 1 = 0.</p><p>Simplifying further, we get -cos^2(x) + cos(x) = 0.</p><p>Now, we can factor out cos(x) from the equation: cos(x)(-cos(x) + 1) = 0.</p><p>To find the solutions, we set each factor equal to zero:</p><ol><li>cos(x) = 0:</li><li>This occurs when x is equal to π/2 or 3π/2 (or any odd multiple of π/2), since cos(x) = 0 at those points.</li><li>-cos(x) + 1 = 0:</li><li>Adding cos(x) to both sides of the equation, we get 1 = cos(x).</li><li>This occurs when x is equal to 0 or 2π (or any even multiple of π), since cos(x) = 1 at those points.</li></ol><p>Therefore, the solutions to the equation sin(x) + cos(x) = 1 are:</p><p>x = π/2 + 2πk, 3π/2 + 2πk, 0 + 2πk, and 2π + 2πk,</p><p>where k is an integer.</p><p>These solutions represent all possible values of x that satisfy the given equation.</p>