Counterexample/Examples/Sum of Sine and Cosine equals 1

Example of Counterexample

Let $P$ be the statement:

$\forall x \in \R: \cos x + \sin x = 1$

A counterexample to $P$ is the real number $\dfrac \pi 4 \in \R$:

\(\ds \cos \dfrac \pi 4\)

\(=\)

\(\ds \dfrac {\sqrt 2} 2\)

Cosine of $\dfrac \pi 4$

\(\ds \sin \dfrac \pi 4\)

\(=\)

\(\ds \dfrac {\sqrt 2} 2\)

Sine of $\dfrac \pi 4$

\(\ds \leadsto \ \ \)

\(\ds \cos \dfrac \pi 4 + \sin \dfrac \pi 4\)

\(=\)

\(\ds \sqrt 2\)

\(\ds \ne 1\)

Sources

• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): counterexample