Evaluate the expression:

$$(\sin heta + \csc heta)^2 + (\cos heta + \sec heta)^2 = ?$$

Question

Evaluate the expression:

$$(\sin 	heta + \csc 	heta)^2 + (\cos 	heta + \sec 	heta)^2 = ?$$

Accepted Answer

**Solution:**

1. Expand the squares using the formula $(a+b)^2 = a^2 + 2ab + b^2$:

$$
   (\sin 	heta + \csc 	heta)^2 + (\cos 	heta + \sec 	heta)^2 = (\sin^2 	heta + 2 \sin 	heta \csc 	heta + \csc^2 	heta) + (\cos^2 	heta + 2 \cos 	heta \sec 	heta + \sec^2 	heta)
   $$

2. Simplify the products using $\csc 	heta = \frac{1}{\sin 	heta}$ and $\sec 	heta = \frac{1}{\cos 	heta}$:

$$
   2 \sin 	heta \csc 	heta = 2 	imes 1 = 2
   $$

$$
   2 \cos 	heta \sec 	heta = 2 	imes 1 = 2
   $$

3. Substitute back:

$$
   (\sin^2 	heta + 2 + \csc^2 	heta) + (\cos^2 	heta + 2 + \sec^2 	heta)
   $$

4. Group terms and use $\sin^2 	heta + \cos^2 	heta = 1$:

$$
   1 + 4 + \csc^2 	heta + \sec^2 	heta = 5 + \csc^2 	heta + \sec^2 	heta
   $$

5. Use the identities $\csc^2 	heta = 1 + \cot^2 	heta$ and $\sec^2 	heta = 1 + 	an^2 	heta$:

$$
   5 + (1 + \cot^2 	heta) + (1 + 	an^2 	heta) = 7 + 	an^2 	heta + \cot^2 	heta
   $$

**Answer:** $7 + 	an^2 	heta + \cot^2 	heta$

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