Primitive of Reciprocal of p by Sine of a x plus q by Cosine of a x/Lemma
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Lemma for Primitive of Reciprocal of p by Sine of a x plus q by Cosine of a x
- $\ds \frac 1 2 \map \arctan {\dfrac {-p} q} + \frac \pi 4 = \frac {\arctan \dfrac q p} 2$
Proof
| \(\ds y\) | \(=\) | \(\ds \frac 1 2 \arctan \dfrac {-p} q + \frac \pi 4\) | making a definition | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 2 y - \frac \pi 2\) | \(=\) | \(\ds \arctan \dfrac {-p} q\) | rearranging | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map \tan {2 y - \frac \pi 2}\) | \(=\) | \(\ds \dfrac {-p} q\) | Definition of Real Arctangent | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map \tan {2 y + \frac {3 \pi} 2}\) | \(=\) | \(\ds \dfrac {-p} q\) | Tangent Function is Periodic on Reals | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds - \map \cot {2 y}\) | \(=\) | \(\ds \dfrac {-p} q\) | Tangent of Angle plus Three Right Angles | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map \tan {2 y}\) | \(=\) | \(\ds \dfrac q p\) | Cotangent is Reciprocal of Tangent | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds \frac {\arctan \dfrac q p} 2\) | Definition of Real Arctangent |
$\blacksquare$