Oefening $$ 6-2 $$ : Trigonometriese identiteite 1. Vereenvoudig elk van die volgende tot een trigonometriese verhouding: a) $$ \frac{\sin \alpha}{ an \alpha} $$ b) $$ \cos ^{2} heta an ^{2} heta+ an ^{2} heta \sin ^{2} heta $$ c) $$ 1-\sin heta \cos heta an heta $$ d) $$ \left(\frac{1-\cos ^{2} \beta}{\cos ^{2} \beta} ight)- an ^{2} \beta $$ 2. Bewys die volgende identiteite en noem die ontoelaatbare waardes of beperkings waar van toepassing: a) $$ \frac{1+\sin heta}{\cos heta}=\frac{\cos heta}{1-\sin heta} $$ b) $$ \sin ^{2} \alpha+(\cos \alpha- an \alpha)(\cos \alpha+ an \alpha)=1- an ^{2} \alpha $$ c) $$ \frac{1}{\cos heta}-\frac{\cos heta an ^{2} heta}{1}=\cos heta $$ d) $$ \frac{2 \sin heta \cos heta}{\sin heta+\cos heta}=\sin heta+\cos heta-\frac{1}{\sin heta+\cos heta} $$ e) $$ \left(\frac{\cos \beta}{\sin \beta}+ an \beta ight) \cos \beta=\frac{1}{\sin \beta} $$ f) $$ \frac{1}{1+\sin heta}+\frac{1}{1-\sin heta}=\frac{2 an heta}{\sin heta \cos heta} $$ g) $$ \frac{\left(1+ an ^{2} \alpha ight) \cos \alpha}{(1- an \alpha)}=\frac{1}{\cos \alpha-\sin \alpha} $$

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Oefening $$ 6-2 $$ : Trigonometriese identiteite 1. Vereenvoudig elk van die volgende tot een trigonometriese verhouding: a) $$ \frac{\sin \alpha}{	an \alpha} $$ b) $$ \cos ^{2} 	heta 	an ^{2} 	heta+	an ^{2} 	heta \sin ^{2} 	heta $$ c) $$ 1-\sin 	heta \cos 	heta 	an 	heta $$ d) $$ \left(\frac{1-\cos ^{2} \beta}{\cos ^{2} \beta}ight)-	an ^{2} \beta $$ 2. Bewys die volgende identiteite en noem die ontoelaatbare waardes of beperkings waar van toepassing: a) $$ \frac{1+\sin 	heta}{\cos 	heta}=\frac{\cos 	heta}{1-\sin 	heta} $$ b) $$ \sin ^{2} \alpha+(\cos \alpha-	an \alpha)(\cos \alpha+	an \alpha)=1-	an ^{2} \alpha $$ c) $$ \frac{1}{\cos 	heta}-\frac{\cos 	heta 	an ^{2} 	heta}{1}=\cos 	heta $$ d) $$ \frac{2 \sin 	heta \cos 	heta}{\sin 	heta+\cos 	heta}=\sin 	heta+\cos 	heta-\frac{1}{\sin 	heta+\cos 	heta} $$ e) $$ \left(\frac{\cos \beta}{\sin \beta}+	an \betaight) \cos \beta=\frac{1}{\sin \beta} $$ f) $$ \frac{1}{1+\sin 	heta}+\frac{1}{1-\sin 	heta}=\frac{2 	an 	heta}{\sin 	heta \cos 	heta} $$ g) $$ \frac{\left(1+	an ^{2} \alphaight) \cos \alpha}{(1-	an \alpha)}=\frac{1}{\cos \alpha-\sin \alpha} $$

UpStudy AI · Accepted Answer

We know that $$ \tan \alpha = \frac{\sin \alpha}{\cos \alpha} $$. Therefore, we can rewrite the expression:
$$
\frac{\sin \alpha}{\tan \alpha} = \frac{\sin \alpha}{\frac{\sin \alpha}{\cos \alpha}} = \cos \alpha
\
We can factor out $$ \tan^{2} \theta $$:
\[
\tan^{2} \theta (\cos^{2} \theta + \sin^{2} \theta)
$$
Using the Pythagorean identity $$ \cos^{2} \theta + \sin^{2} \theta = 1 $$:
$$
\tan^{2} \theta \cdot 1 = \tan^{2} \theta
\
We know that $$ \tan \theta = \frac{\sin \theta}{\cos \theta} $$:
\[
1 - \sin \theta \cdot \frac{\sin \theta}{\cos \theta} = 1 - \frac{\sin^{2} \theta}{\cos \theta}
$$
This cannot be simplified further without specific values.
Using the identity $$ 1 - \cos^{2} \beta = \sin^{2} \beta $$:
$$
\frac{\sin^{2} \beta}{\cos^{2} \beta} - \tan^{2} \beta = \tan^{2} \beta - \tan^{2} \beta = 0
\### Part 2: Proving Trigonometric Identities
Cross-multiplying gives:
\[
(1 + \sin \theta)(1 - \sin \theta) = \cos^{2} \theta
$$
Expanding the left side:
$$
1 - \sin^{2} \theta = \cos^{2} \theta
$$
This is true by the Pythagorean identity. The restrictions are $$ \cos \theta \neq 0 $$ and $$ \sin \theta \neq 1 $$.
Expanding the left side:
$$
\sin^{2} \alpha + \cos^{2} \alpha - \tan^{2} \alpha = 1 - \tan^{2} \alpha
$$
This is true by the Pythagorean identity. The restriction is $$ \cos \alpha \neq 0 $$.
Rewriting $$ \tan^{2} \theta $$:
$$
\frac{1}{\cos \theta} - \frac{\sin^{2} \theta}{\cos \theta} = \frac{1 - \sin^{2} \theta}{\cos \theta} = \frac{\cos^{2} \theta}{\cos \theta} = \cos \theta
$$
The restriction is $$ \cos \theta \neq 0 $$.
Cross-multiplying gives:
$$
2 \sin \theta \cos \theta = (\sin \theta + \cos \theta)^{2} - 1
$$
Expanding the right side:
$$
2 \sin \theta \cos \theta = \sin^{2} \theta + 2 \sin \theta \cos \theta + \cos^{2} \theta - 1
$$
This simplifies to $$ 1 - 1 = 0 $$, which is true. The restriction is $$ \sin \theta + \cos \theta \neq 0 $$.
Rewriting $$ \tan \beta $$:
$$
\left(\frac{\cos \beta}{\sin \beta} + \frac{\sin \beta}{\cos \beta}\right) \cos \beta = \frac{\cos^{2} \beta + \sin^{2} \beta}{\sin \beta} = \frac{1}{\sin \beta}
$$
This is true by the Pythagorean identity. The restriction is $$ \sin \beta \neq 0 $$.
Finding a common denominator on the left side:
$$
\frac{(1 - \sin \theta) + (1 + \sin \theta)}{(1 + \sin \theta)(1 - \sin \theta)} = \frac{2}{1 - \sin^{2} \theta} = \frac{2}{\cos^{2} \theta}
$$
This simplifies to $$ 2 \sec^{2} \theta $$, which is equal to $$ \frac{2 \tan \theta}{\sin \theta \cos \theta} $$. The restriction is $$ \sin \theta \neq \pm 1 $$.
**Solutions:** 1. **Simplifying Trigonometric Expressions:** - a) $$ \cos \alpha $$ - b) $$ \tan^{2} \theta $$ - c) $$ 1 - \frac{\sin^{2} \theta}{\cos \theta} $$ - d) $$ 0 $$ 2. **Proved Trigonometric Identities with Restrictions:** - a) $$ \frac{1+\sin \theta}{\cos \theta} = \frac{\cos \theta}{1-\sin \theta} $$ (Restriction: $$ \cos \theta \neq 0 $$ and $$ \sin \theta \neq 1 $$) - b) $$ \sin^{2} \alpha + (\cos \alpha - \tan \alpha)(\cos \alpha + \tan \alpha) = 1 - \tan^{2} \alpha $$ (Restriction: $$ \cos \alpha \neq 0 $$) - c) $$ \frac{1}{\cos \theta} - \frac{\cos \theta \tan^{2} \theta}{1} = \cos \theta $$ (Restriction: $$ \cos \theta \neq 0 $$) - d) $$ \frac{2 \sin \theta \cos \theta}{\sin \theta + \cos \theta} = \sin \theta + \cos \theta - \frac{1}{\sin \theta + \cos \theta} $$ (Restriction: $$ \sin \theta + \cos \theta \neq 0 $$) - e) $$ \left(\frac{\cos \beta}{\sin \beta} + \tan \beta\right) \cos \beta = \frac{1}{\sin \beta} $$ (Restriction: $$ \sin \beta \neq 0 $$) - f) $$ \frac{1}{1+\sin \theta} + \frac{1}{1-\sin \theta} = \frac{2 \tan \theta}{\sin \theta \cos \theta} $$ (Restriction: $$ \sin \theta \neq \pm 1 $$) - g) $$ \frac{(1+\tan^{2} \alpha) \cos \alpha}{(1-\tan \alpha)} = \frac{1}{\cos \alpha - \sin \alpha} $$ (Restriction: $$ \cos \alpha \neq \sin \alpha $$)

Oefening : Trigonometriese identiteite

Vereenvoudig elk van die volgende tot een trigonometriese verhouding:
a)
b)
c)
d)

Bewys die volgende identiteite en noem die ontoelaatbare waardes of beperkings waar van toepassing:
a)
b)
c)
d)
e)
f)
g)

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Oefening : Trigonometriese identiteite Vereenvoudig elk van die volgende tot een trigonometriese verhouding: a) b) c) d) Bewys die volgende identiteite en noem die ontoelaatbare waardes of beperkings waar van toepassing: a) b) c) d) e) f) g)