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If
sin A=p and
cos A=q :
5.3.1 Write
tan A in terms of
p and
q
5.3.2 Simplify
p^(4)-q^(4) to a single trigonometric ratio

If $\sin \mathrm{A}=p$ and $\cos \mathrm{A}=q$ : $\newline$ $5$ . $3$ . $1$ Write $\tan \mathrm{A}$ in terms of $p$ and $q$ $\newline$ $5$ . $3$ . $2$ Simplify $p^{4}-q^{4}$ to a single trigonometric ratio

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Write inverse variation equations

Full solution

Q. If

\sin \mathrm{A}=p

and

\cos \mathrm{A}=q

:

\newline

5

.

3

.

1

Write

\tan \mathrm{A}

in terms of

p

and

q

\newline

5

.

3

.

2

Simplify

p^{4}-q^{4}

to a single trigonometric ratio

Trigonometric Identity: $\tan A = \frac{\sin A}{\cos A}$ $\tan A = \frac{p}{q}$
Factoring Difference of Squares: $p^4 - q^4$ can be factored as a difference of squares. $\newline$ $(p^2)^2 - (q^2)^2 = (p^2 + q^2)(p^2 - q^2)$
Pythagorean Identity: We know that $\sin^2 A + \cos^2 A = 1$ . $\newline$ So, $p^2 + q^2 = 1$ .
Substitution: Substitute $1$ for $p^2 + q^2$ in the factored form. $\newline$ $(1)(p^2 - q^2)$
Simplify Using Pythagorean Identity: Now, we simplify $p^2 - q^2$ using the Pythagorean identity. $\newline$ $p^2 - q^2 = \sin^2 A - \cos^2 A$
Double Angle Formula: $\sin^2 A - \cos^2 A$ can be written as $\cos(2A)$ using the double angle formula for cosine. $\newline$ $\cos(2A) = \cos^2 A - \sin^2 A$ $\newline$ But we need $\sin^2 A - \cos^2 A$ , which is $-\cos(2A)$ .
Final Simplification: So, $p^4 - q^4$ simplifies to $-\cos(2A)$ .

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