Q.Find the general solution of the equation $5\cos^2\theta + 7\sin^2\theta - 6 = 0$.

The key idea is to transform the given trigonometric equation into a simpler form involving a single trigonometric function, then apply the standard general solution formula for that form.

First, rewrite $\cos^2\theta$ using the identity $\cos^2\theta = 1 - \sin^2\theta$:

$5(1 - \sin^2\theta) + 7\sin^2\theta - 6 = 0$

$5 - 5\sin^2\theta + 7\sin^2\theta - 6 = 0$

Combine terms to simplify the equation:

$2\sin^2\theta - 1 = 0$

$2\sin^2\theta = 1$

$\sin^2\theta = \frac{1}{2}$

Recognize that $\frac{1}{2}$ is $\sin^2(\frac{\pi}{4})$, since $\sin(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}$.

So, $\sin^2\theta = \sin^2\left(\frac{\pi}{4}\right)$. …

The problem asks for the general solution of a trigonometric equation. Our primary goal in such problems is to simplify the given equation into one of the standard forms for which we know the general solution. These standard forms typically involve a single trigonometric function, like $\sin x = \sin y$, $\cos x = \cos y$, or $\tan x = \tan y$.

The given equation, $5\cos^2\theta + 7\sin^2\theta - 6 = 0$, contains both $\sin^2\theta$ and $\cos^2\theta$. The most straightforward way to reduce this to a single trigonometric function is to use the fundamental trigonometric identity:

From this identity, we can express $\cos^2\theta$ as $1 - \sin^2\theta$ (or $\sin^2\theta$ as $1 - \cos^2\theta$). Substituting either of these into the original equation will allow us to work with only one trigonometric function.

Let's proceed with the steps:

1. Convert to a single trigonometric function. We will use the identity $\cos^2\theta = 1 - \sin^2\theta$ to express the entire equation in terms of $\sin^2\theta$. Substitute this into the given equation:

Now, expand and simplify the equation:

Combine the constant terms and the $\sin^2\theta$ terms:

2. Solve for the trigonometric function. From the simplified equation, isolate $\sin^2\theta$:

3. Express the constant term in the form of the squared trigonometric function. We need to find an angle $\alpha$ such that $\sin^2\alpha = \frac{1}{2}$. We know that $\sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}$. Therefore, $\sin^2\left(\frac{\pi}{4}\right) = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2}$. So, our equation becomes:

4. Apply the general solution formula. We use the standard general solution formula for equations of the form $\sin^2 x = \sin^2 \alpha$. …