# (SOLVED) Simplify $\cos^{-1}\left( \frac{7}{2}\left( 1+\cos{2x} \right) + \sqrt{\left( \sin^{2}{x} - 48\cos^{2}{x} \right)}\sin{x}\right)$

I need a little help in simplification of an ITF problem of a college entrance exam I got stuck in.

$$\cos^{-1}\left( \frac{7}{2}\left( 1+\cos{2x} \right) + \sqrt{\left( \sin^{2}{x} - 48\cos^{2}{x} \right)}\sin{x}\right)$$

with
$$x\in \left( 0,\frac{\pi}{2} \right)$$

What I'm doing:

$$y=\cos ^{-1}\left(\frac{7}{2}(1+\cos 2 x)+\sqrt{\left(\sin ^2 x-48 \cos ^2 x\right)} \sin x\right)$$

$$\quad=\cos ^{-1}\left((7 \cos x)(\cos x)+\sqrt{1-49 \cos ^2 x} \sqrt{1-\cos ^2 x}\right)$$ ......stuck

Thanks!

• Please consider commenting if there is something to improve instead of downvoting recklessly. Not everyone is 40 years old with 20 years of experience with maths. Some of us are in high school and get stuck sometimes. Commented May 11, 2023 at 11:26
• In fact, it cannot be right, unless you want to wade into complex numbers. The square root, for small values of $x$, will not have a positive argument because $\sin x (\sin^2 x - 48 \cos^2 x) < 0$ for small positive values of $x$. Commented May 11, 2023 at 11:32
• It should be $\sqrt{\left( \sin^{2}{x} - 48\cos^{2}{x} \right)}\,\sin{x}$ not $\sqrt{\left( \sin^{2}{x} - 48\cos^{2}{x} \right)\sin{x}}$. Commented May 11, 2023 at 11:35
• Certainly not much point answering the question if we are not sure what the formula should look like. One misplaced symbol here may turn an easy problem into an unsolvable one!
– user700480
Commented May 11, 2023 at 11:39
• @44yu5h are you familiar with the formula: $\arccos x-\arccos y$? Commented May 11, 2023 at 11:58

Suppose $$\cos^{-1}\frac17\le x\le\frac\pi2$$, and take $$\theta$$ in $$[0,\frac\pi2]$$ such that $$\cos\theta=7\cos x$$. Then \eqalign{ \frac{7}{2}(1+\cos 2 x)+{}&\sqrt{\left(\sin ^2 x-48 \cos ^2 x\right)} \sin x\cr &=(7 \cos x)(\cos x)+\sqrt{1-49 \cos ^2 x}\sin x\cr &=\cos x\cos\theta+\sin x\sin\theta\cr &=\cos(x-\theta)\cr} and so $$\cos^{-1}({\rm this})=x-\theta=x-\cos^{-1}(7\cos x)\ ,$$ provided $$0\le x-\cos^{-1}(7\cos x)\le\pi$$, and this is true for all $$x$$ in the range stated above.