Fundamental Trigonometric Identities: Level 4 Challenges Practice Problems Online

Evaluate the sum

$\begin{aligned} & \ \ \ \log_{\cos1}(\tan{1}) \\ &+\log_{\cos{2}}(\tan{2}) \\ &+ \log_{\cos{3}}(\tan{3}) \\& +\ldots \\ &+ \log_{\cos{44}} (\tan{44}) \\ &+ \log_{\sin{45}}(\tan{45}) \\ &+ \log_{\sin46}(\tan{46}) \\ &+\ldots \\ &+ \log_{\sin89}(\tan{89}). \end{aligned}$

Note: All angles are in degrees, and be aware that the base changes from $\cos$ to $\sin$ .

$\large (7\cos x+24\sin x)(7\sin x-24\cos x)$

Find the maximum value of this expression over all real values $x.$

Hint:

If $x$ and $y$ are acute angles such that

$\frac {\sin x}{\sin y } = \frac {1}{2}, \quad \frac {\cos x}{\cos y } = \frac 3 2 ,$

what is $\tan^2 (x+y)?$

Define the function $f(x)=\frac{2x}{1-x^2}$ . Find the number of distinct real solutions of the equation $f^{(5)} (x) =x.$

Details and assumptions

$f^{(n)} (x)$ denotes the function $f$ applied $n$ times. In particular, $f^{(5)} (x) = f(f(f(f(f(x)))))$ .

Find $\cos{1˚} \cos{2˚}+\cos{2˚} \cos{3˚}+ \cdots +\cos{88˚} \cos{89˚}.$

Give your answer to two decimal places.

Concept Quizzes

Challenge Quizzes

Fundamental Trigonometric Identities: Level 4 Challenges

Excel in math and science

Master concepts by solving fun, challenging problems.

It's hard to learn from lectures and videos

Learn more effectively through short, interactive explorations.

Used and loved by over 7 million people

Learn from a vibrant community of students and enthusiasts, including olympiad champions, researchers, and professionals.