Prove the fulaving Identities Q1. $$ \sin heta \cot heta \equiv \cos heta $$ Q2. $$ \cos ^{2} heta-\sin ^{2} heta=1-25 \sin ^{2} heta $$ 43. $$ \cot ^{2} heta\left(1-\cos ^{2} heta ight)=\cos ^{2} heta $$

Question

Prove the fulaving Identities Q1. $$ \sin 	heta \cot 	heta \equiv \cos 	heta $$ Q2. $$ \cos ^{2} 	heta-\sin ^{2} 	heta=1-25 \sin ^{2} 	heta $$ 43. $$ \cot ^{2} 	heta\left(1-\cos ^{2} 	hetaight)=\cos ^{2} 	heta $$

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**Q1. Prove that**  
$$
\sin \theta \cot \theta \equiv \cos \theta
$$

1. Recall that by definition,  
   $$
   \cot \theta = \frac{\cos \theta}{\sin \theta}
   $$
2. Substitute the definition into the left-hand side:  
   $$
   \sin \theta \cot \theta = \sin \theta \cdot \frac{\cos \theta}{\sin \theta}
   $$
3. Cancel $$\sin \theta$$ (assuming $$\sin \theta \neq 0$$):  
   $$
   \sin \theta \cdot \frac{\cos \theta}{\sin \theta} = \cos \theta
   $$
4. Thus, the identity is verified:  
   $$
   \sin \theta \cot \theta \equiv \cos \theta.
   $$

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**Q2. Investigate whether the identity**  
$$
\cos^2 \theta - \sin^2 \theta = 1 - 25 \sin^2 \theta
$$
**holds for all $$\theta$$.**

1. Begin with the known Pythagorean identity:  
   $$
   \cos^2 \theta = 1 - \sin^2 \theta.
   $$
2. Substitute into the left-hand side:  
   $$
   \cos^2 \theta - \sin^2 \theta = (1 - \sin^2 \theta) - \sin^2 \theta = 1 - 2\sin^2 \theta.
   $$
3. Equate this to the right-hand side:  
   $$
   1 - 2\sin^2 \theta = 1 - 25\sin^2 \theta.
   $$
4. Subtract $$1$$ from both sides:  
   $$
   -2\sin^2 \theta = -25\sin^2 \theta.
   $$
5. Multiply both sides by $$-1$$:  
   $$
   2\sin^2 \theta = 25\sin^2 \theta.
   $$
6. For a general angle, unless $$\sin^2 \theta = 0$$, we can divide both sides by $$\sin^2 \theta$$:  
   $$
   2 = 25,
   $$
   which is a contradiction.
7. Therefore, the equality $$ \cos^2 \theta - \sin^2 \theta = 1 - 25\sin^2 \theta $$ holds only when  
   $$
   \sin^2 \theta = 0 \quad \text{(i.e. when $$\sin \theta = 0$$)}.
   $$
8. In conclusion, the given expression is not an identity valid for all $$\theta$$; it holds only for those angles satisfying $$\sin \theta = 0$$.

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**Q3. Prove that**  
$$
\cot^2 \theta \left( 1 - \cos^2 \theta \right) = \cos^2 \theta.
$$

1. Recognize that  
   $$
   1 - \cos^2 \theta = \sin^2 \theta,
   $$
   by the Pythagorean identity.
2. Substitute this into the left-hand side:  
   $$
   \cot^2 \theta \left( 1 - \cos^2 \theta \right) = \cot^2 \theta \sin^2 \theta.
   $$
3. Recall the definition of $$\cot \theta$$:  
   $$
   \cot \theta = \frac{\cos \theta}{\sin \theta},
   $$
   so that  
   $$
   \cot^2 \theta = \left(\frac{\cos \theta}{\sin \theta}\right)^2 = \frac{\cos^2 \theta}{\sin^2 \theta}.
   $$
4. Substitute this into the expression:  
   $$
   \cot^2 \theta \sin^2 \theta = \frac{\cos^2 \theta}{\sin^2 \theta} \cdot \sin^2 \theta.
   $$
5. Cancel $$\sin^2 \theta$$ (assuming $$\sin \theta \neq 0$$):  
   $$
   \frac{\cos^2 \theta}{\sin^2 \theta} \cdot \sin^2 \theta = \cos^2 \theta.
   $$
6. Hence, the identity is established:  
   $$
   \cot^2 \theta \left( 1 - \cos^2 \theta \right) = \cos^2 \theta.
   $$
**Q1.** $$ \sin \theta \cot \theta = \cos \theta $$ **Q2.** The identity $$ \cos^2 \theta - \sin^2 \theta = 1 - 25 \sin^2 \theta $$ holds only when $$\sin \theta = 0$$. **Q3.** $$ \cot^2 \theta (1 - \cos^2 \theta) = \cos^2 \theta $$

Prove the fulaving
Identities
Q1.
Q2.
43.

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Prove the fulaving Identities Q1. Q2. 43.