If two trigonometric expressions are defined as x = 8 cos θ + 15 sin θ and y = 8 sin θ - 15 cos θ, find the value of $x^2+y^2$.

The correct answer is 289.

Given the two trigonometric expressions:

$x=8cos\theta +15sin\theta$

$y=8sin\theta -15cos\theta$

We are asked to find the value of $x^2+y^2$.

First, calculate $x^2$ by squaring both sides of the expression for $x$:

$x^2={(8cos\theta +15sin\theta )}^2$

Applying the formula ${(a+b)}^2=a^2+b^2+2ab$:

$x^2=64{cos}^2\theta +225{sin}^2\theta +240sin\theta cos\theta$

Next, calculate $y^2$ by squaring both sides of the expression for $y$:

$y^2={(8sin\theta -15cos\theta )}^2$

Applying the formula ${(a-b)}^2=a^2+b^2-2ab$:

$y^2=64{sin}^2\theta +225{cos}^2\theta -240sin\theta cos\theta$

Now, add the expressions for $x^2$ and $y^2$:

$x^2+y^2=(64{cos}^2\theta +225{sin}^2\theta +240sin\theta cos\theta )+(64{sin}^2\theta +225{cos}^2\theta -240sin\theta cos\theta )$

Notice that the middle terms $240sin\theta cos\theta$ and $-240sin\theta cos\theta$ cancel each other out:

$x^2+y^2=64({cos}^2\theta +{sin}^2\theta )+225({sin}^2\theta +{cos}^2\theta )$

Using the fundamental trigonometric identity ${sin}^2\theta +{cos}^2\theta =1$:

$x^2+y^2=64\left(1\right)+225\left(1\right)$

$x^2+y^2=64+225=289$

Therefore, $x^2+y^2=289$.