Trigonometric Equation Solution

We are given the equation: $10\cos^2\theta + 7\sin^2\theta = 7$ The angle constraint is $0^\circ \le \theta \le 90^\circ$.

To solve for $\theta$, we can use the trigonometric identity $\cos^2\theta = 1 - \sin^2\theta$. Substitute this into the equation: $10(1 - \sin^2\theta) + 7\sin^2\theta = 7$

Expand the term: $10 - 10\sin^2\theta + 7\sin^2\theta = 7$

Combine the terms involving $\sin^2\theta$: $10 - 3\sin^2\theta = 7$

Rearrange the equation to isolate $\sin^2\theta$: $3\sin^2\theta = 10 - 7$ $3\sin^2\theta = 3$

Solve for $\sin^2\theta$: $\sin^2\theta = \frac{3}{3}$ $\sin^2\theta = 1$

Given that $0^\circ \le \theta \le 90^\circ$, the value of $\sin\theta$ must be non-negative. Taking the square root of both sides: $\sin\theta = \sqrt{1}$ $\sin\theta = 1$

The angle $\theta$ for which $\sin\theta = 1$ in the specified range is $\theta = 90^\circ$.

Now, find the value of $\cos\theta$ for $\theta = 90^\circ$: $\cos\theta = \cos(90^\circ) = 0$

Calculate $(\sin\theta + \cos\theta)$

We need to find the value of the expression $(\sin\theta + \cos\theta)$.

Substitute the determined values of $\sin\theta$ and $\cos\theta$: $(\sin\theta + \cos\theta) = (1 + 0)$ $(\sin\theta + \cos\theta) = 1$