Prove \(\sin\theta\cot\theta=\cos\theta\)

Prove \(\tan x+\cos x=\sin x(\sec x+\cot x)\).

Prove \(\displaystyle\frac{\cos^4t-\sin^4t}{\cos^2t}=1-\tan^2t\).

Prove \(1+\cos\theta=\displaystyle\frac{\sin^2\theta}{1-\cos\theta}\).

Prove \(\tan x+\cot x=\sec x\csc x\).

Prove \(\displaystyle\frac{\sin\alpha}{1+\cos\alpha}+\displaystyle\frac{1+\cos\alpha}{\sin\alpha}=2\csc\alpha\).

Prove \(\displaystyle\frac{1+\sin t}{\cos t}=\displaystyle\frac{\cos t}{1-\sin t}\).

Show that \(\cot^2\theta+\cos^2\theta=\cot^2\theta\cos^2\theta\) is not an identity by finding a counterexample.

Mini Lecture
Prove.

1. \(\cos\theta\tan\theta=\sin\theta\)

2. \(\cos x(\csc x+\tan x)=\cot x+\sin x\)

3. \(\displaystyle\frac{\cos^4t-\sin^4t}{\sin^2t}=\cot^2t-1\)

Find the exact value for \(\cos 75^\circ\).

Show that \(\cos(x+2\pi)=\cos x\).

Write \(\cos 3x\cos 2x-\sin 3x\sin 2x\) as a single cosine.

Show that \(\cos(90^\circ -A)=\sin A\).

Find the exact value of \(\sin \left(\displaystyle\frac{\pi}{12}\right)\).

If \(\sin A=\frac{3}{5}\) with \(A\) in QI and \(\cos B=-\frac{5}{13}\) with \(B\) in QIII, find \(\sin(A+B)\), \(\cos(A+B)\), and \(\tan(A+B)\).

If \(\sin A=\frac{3}{5}\) with \(A\) in QI and \(\cos B=-\frac{5}{13}\) with \(B\) in QIII, find \(\tan(A+B)\) by using this formula \[\tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}\]

Mini Lecture

1. Find the exact value of \(\sin 15^\circ\).

2. Prove: \(\sin(90^\circ +\theta)=\cos\theta\)

3. Graph one cycle: \(y=2\left(\sin x\cos\frac{\pi}{3}-\cos x\sin\frac{\pi}{3}-\cos x\sin\frac{\pi}{3}\right)\)

If \(\sin A=\displaystyle\frac{3}{5}\) with \(A\) in QII, find \(\sin 2A\).

Prove \(\left(\sin\theta+\cos\theta\right)^2=1+\sin 2\theta\).

If \(\sin A=\displaystyle\frac{1}{\sqrt{5}}\), find \(\cos 2A\).

Prove \(\sin 2x=\displaystyle\frac{2\cot x}{\left(1+\cot^2x\right)}\).

Prove \(\cos 4x=8\cos^4x-8\cos^2x+1\).

Prove \(\tan\theta=\displaystyle\frac{1-\cos2\theta}{\sin2\theta}\).

Graph \(y=3-6\sin^2x\) from \(x=0\) to \(x=2\pi\).

Mini Lecture

1. If \(\cos x=\displaystyle\frac{1}{\sqrt{10}}\) with \(x\in\) QIV, find \(\sin 2x\).

2. Graph \(y=4-8\sin^2x\) from \(0\leq x\leq 2\pi\)

3. Verify \(\sin 60^\circ=2\sin30^\circ\cos30^\circ\)

If \(\cos A=\frac{3}{5}\) with \(270^\circ<A<360^\circ\), find \(\sin\frac{A}{2}\), \(\cos\frac{A}{2}\), and \(\tan\frac{A}{2}\).

If \(\sin A=-\frac{12}{13}\) with \(180^\circ<A<270^\circ\), find the six trigonometric functions of \(\displaystyle\frac{A}{2}\).

Find \(\tan15^\circ\).

Prove \(\sin^2\displaystyle\frac{x}{2}=\displaystyle\frac{\tan x-\sin x}{2\tan x}\).

Mini Lecture
Let \(\sin A=\displaystyle\frac{4}{5}\) with \(A\) in QII. Find.

1. \(\sin\displaystyle\frac{A}{2}\)

2. \(\cos 2A\)

3. \(\sec 2A\)

Evaluate \(\sin\left(\arcsin\displaystyle\frac{3}{5}+\arctan 2\right)\) without using a calculator.

Write \(\sin\left(2\tan^{-1}x\right)\) as an equivalent expression involving only \(x\). (Assume \(x\) is positive).

Verify the product formula \((3)\) for \(A=30^\circ\) and \(B=120^\circ\).

Write \(10\cos5x\sin3x\) as a sum or difference.

Verify sum formula \((7)\) for \(\alpha=30^\circ\) and \(\beta=90^\circ\).

Verify the identity \(-\tan x=\displaystyle\frac{\cos 3x-\cos x}{\sin 3x+\sin x}\).

Mini Lecture
Evaluate.

1. \(\sin\left(\arcsin\displaystyle\frac{3}{5}-\arctan2\right)\)

2. \(\tan\left(\sin^{-1}x\right)\)

3. Rewrite as a product: \(\sin7x+\sin3x\)