Proving Trig Identities Prove the following using trigonometric identities. $$ \begin{array}{lll} ext { 44. } \frac{ an ^{2} x}{\sin ^{2} x}=\frac{1}{\cos ^{2} x} & ext { 45. } \sec ^{2} x+\csc ^{2} x=\frac{\csc ^{2} x}{\cos ^{2} x}\end{array} $$

Question

Proving Trig Identities Prove the following using trigonometric identities. $$ \begin{array}{lll}	ext { 44. } \frac{	an ^{2} x}{\sin ^{2} x}=\frac{1}{\cos ^{2} x} & 	ext { 45. } \sec ^{2} x+\csc ^{2} x=\frac{\csc ^{2} x}{\cos ^{2} x}\end{array} $$

UpStudy AI · Accepted Answer

Certainly! Let's prove each trigonometric identity step by step using fundamental trigonometric identities.

---

### **Problem 44**

**Prove**:
$$
\frac{\tan^2 x}{\sin^2 x} = \frac{1}{\cos^2 x}
$$

**Proof**:

1. **Express $$\tan^2 x$$ in terms of sine and cosine**:
   $$
   \tan x = \frac{\sin x}{\cos x} \implies \tan^2 x = \left(\frac{\sin x}{\cos x}\right)^2 = \frac{\sin^2 x}{\cos^2 x}
   $$

2. **Substitute $$\tan^2 x$$ into the left-hand side (LHS)**:
   $$
   \frac{\tan^2 x}{\sin^2 x} = \frac{\frac{\sin^2 x}{\cos^2 x}}{\sin^2 x}
   $$

3. **Simplify the expression**:
   $$
   \frac{\frac{\sin^2 x}{\cos^2 x}}{\sin^2 x} = \frac{\sin^2 x}{\cos^2 x} \times \frac{1}{\sin^2 x} = \frac{1}{\cos^2 x}
   $$

4. **Conclusion**:
   $$
   \frac{\tan^2 x}{\sin^2 x} = \frac{1}{\cos^2 x} \quad \text{✓}
   $$

---

### **Problem 45**

**Prove**:
$$
\sec^2 x + \csc^2 x = \frac{\csc^2 x}{\cos^2 x}
$$

**Proof**:

1. **Express $$\sec^2 x$$ and $$\csc^2 x$$ in terms of sine and cosine**:
   $$
   \sec x = \frac{1}{\cos x} \implies \sec^2 x = \frac{1}{\cos^2 x}
   $$
   $$
   \csc x = \frac{1}{\sin x} \implies \csc^2 x = \frac{1}{\sin^2 x}
   $$

2. **Substitute into the left-hand side (LHS)**:
   $$
   \sec^2 x + \csc^2 x = \frac{1}{\cos^2 x} + \frac{1}{\sin^2 x}
   $$

3. **Combine the terms over a common denominator**:
   $$
   \frac{1}{\cos^2 x} + \frac{1}{\sin^2 x} = \frac{\sin^2 x + \cos^2 x}{\sin^2 x \cos^2 x}
   $$

4. **Use the Pythagorean identity** $$\sin^2 x + \cos^2 x = 1$$:
   $$
   \frac{\sin^2 x + \cos^2 x}{\sin^2 x \cos^2 x} = \frac{1}{\sin^2 x \cos^2 x}
   $$

5. **Express the right-hand side (RHS)**:
   $$
   \frac{\csc^2 x}{\cos^2 x} = \frac{\frac{1}{\sin^2 x}}{\cos^2 x} = \frac{1}{\sin^2 x \cos^2 x}
   $$

6. **Conclusion**:
   $$
   \sec^2 x + \csc^2 x = \frac{\csc^2 x}{\cos^2 x} \quad \text{✓}
   $$

---

Both identities have been successfully proven using fundamental trigonometric identities.
**Proven Identities:** 1. $$ \frac{\tan^2 x}{\sin^2 x} = \frac{1}{\cos^2 x} $$ 2. $$ \sec^2 x + \csc^2 x = \frac{\csc^2 x}{\cos^2 x} $$ Both identities have been verified using basic trigonometric relationships.

Proving Trig Identities
Prove the following using trigonometric identities.

Upstudy AI Solution

Answer

Solution

Mind Expander

Related Questions

Latest Trigonometry Questions