Precalculus Solutions | Expert Assistance for Problems & Examples

How is the trig identity $\tan^{2} (x) = \sec^{2} (x) - 1$ derived?

Ethen Wong 2022-01-26

Solving $\frac{d y}{d θ} + y \cos θ = \frac{12}{\sin 2} θ$ with $y (\frac{π}{2}) = 4$

Fiona Petersen 2022-01-26

If $\cos^{4} x . \sec^{2} y, \frac{1}{2}, \sin^{4} x . \csc^{2} y$ are in A.P, then prove that $\cos^{8} x . \sec^{6} y, \frac{1}{2}, \sin^{8} x . \csc^{6} y$ in A.P
My Attempt
$\cos^{4} x . \sec^{2} y + \sin^{4} . x \csc^{2} y = \frac{\cos^{4} x}{\cos^{2} y} + \frac{\sin^{4} x}{\sin^{2} y} = 1$
$\Rightarrow \sin^{2} y . \cos^{4} x + \cos^{2} y . \sin^{4} x = \sin^{2} y . \cos^{2} y$
$\cos^{8} x . \sec^{6} y + \sin^{8} x . \csc^{6} y = \frac{\cos^{8} x}{\cos^{6} y} + \frac{\sin^{8} x}{\sin^{6} y}$
How do I know that the given terms are in A.P, G.P or H.P ?

mairie0708zl 2022-01-26

Proof that $| \arctan (x) - \frac{π}{4} - \frac{(x - 1)}{2} | \leq \frac{{(x - 1)}^{2}}{2}$
I'm trying to prove that, for every $x \geq 1$ :
$| \arctan (x) - \frac{π}{4} - \frac{(x - 1)}{2} | \leq \frac{{(x - 1)}^{2}}{2}$
I could do it graphically on $R$ , but how to make a formal algebraic proof?

Addison Gross 2022-01-25

Proof that $\sum_{n = 1}^{\infty} \frac{{(- 1)}^{n + 1} \sin (n)}{n} = \frac{1}{2}$

veceriraby 2022-01-25

If $\sin a + \sin c = 2 \sin b$ , show that $\tan \frac{a + b}{2} + \tan \frac{b + c}{2} = 2 \tan \frac{c + a}{2}$

Babenzgs 2022-01-25

Separate equations of the pair of lines $x^{2} + 2 x y \sec θ + y^{2} = 0$

Jamya Elliott 2022-01-25

If $a + b + c = 90^{\circ}$ , prove $\tan (a) \cdot \tan (b) + \tan (b) \cdot \tan (c) + \tan (c) \cdot \tan (a) = 1$

Aubrey Hendricks 2022-01-25

If $π < α < \frac{3 π}{2}$ then the expression $\sqrt{4 \sin^{4} α + \sin^{2} 2 α} + 4 \cos^{2} (\frac{π}{4} - \frac{α}{2})$

Marquis Neal 2022-01-25

What did I get wrong when solving $\int \frac{\sqrt{x^{2} - 1}}{x^{4}} d x$ ?
Let $x = \sec θ$
Then $d x = \sec θ \tan θ d θ$
$\int \frac{\sqrt{\sec^{2} θ - 1}}{\sec^{4} θ} \sec θ \tan θ d θ$
$= \int \frac{\sec θ}{\sec^{4} θ} \sqrt{\tan^{2} θ} \tan θ d θ$
$= \int \frac{1}{\sec^{3} θ} \tan^{2} θ d θ$
$= \int \frac{1}{\sec^{3} θ} \frac{\sec^{2} θ}{\csc^{2} θ} d θ$
$= \int \frac{1}{\sec θ} \frac{1}{\csc^{2} θ} d θ$
$= \int \cos θ \sin^{2} θ d θ$
Using u-substition, let $u = \sin θ$
Then $d u = \cos θ d θ and d x = \frac{1}{\cos θ} d u$
$\int \cos θ u^{2} \frac{1}{\cos θ} d u$
$= \int u^{2} d u$
$= \frac{u^{3}}{3} + C$
$= \frac{\sin^{3} θ}{3} + C$
Since $x = \sec θ, \sec^{- 1} (x) = θ$
$= \frac{\sin^{3} (\sec^{- 1} (x))}{3} + C$

Emerson Barnes 2022-01-25

Let $f (x) = 7 x^{2} + 5$ and $g (x) = x - 3$ , how do you find the composite function $(f \circ g) (2)$ ?

Alvin Pugh 2022-01-25

Solving $\sin 3 θ = \frac{12}{}$ on the interval $[0, 2 π]$ . I don't understand where some solutions came from.

Laney Spears 2022-01-25

Solving $lim_{x \to 0} \frac{1}{x^{2}} - \cot^{2} (x)$
My approach :
$lim_{x \to 0} \frac{1}{x^{2}} - \frac{\cos^{2} x}{\sin^{2} x}$
$lim_{x \to 0} \frac{\sin^{2} (x) - x^{2} \cos^{2} (x)}{x^{2} \sin^{2} (x)}$

tripplek7i 2022-01-25

Determine the polar form of $\sin θ + i (1 + \cos θ)$

Journey Stout 2022-01-25

Proving $\arctan (\sqrt{x^{2} + 1}) = \frac{π}{4} - \frac{1}{2} \arctan (x)$

treetopssan 2022-01-25

Solve equation $1 + \sin^{2} θ = 3 \sin θ \cos θ (given {\tan 71}^{\circ} 34' = 3)$

Jazmin Perry 2022-01-25

Evaluate
$\int \frac{d x}{\sin x - \cos x}$
I know it can be done by Weierstrass substitution. But I am looking for new/simple approach. For example I tried:
$\int \frac{1}{\sin x - \cos x} \cdot \frac{\sin x + \cos x}{\sin x + \cos x} d x = \int \frac{\sin x + \cos x}{- \cos 2 x} d x$
but I can't continue from here.

Jaeden Frederick 2022-01-25

I cannot find how or why this,
$5 \sin 3 x - 1 = 3$
Has one of two solutions being this,
$42.99 + n \times 120^{\circ}$
I am lost on how to get this solution. I have found the other one, so I will not mention it.

taoigas 2022-01-25

How do you simplify $P_{5}^{2}$ ?

Sereinserenormg 2022-01-25

How do you write $y = 3 \sqrt{1 + x^{2}}$ as a composition of two simpler functions?

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