If x is an acute angle such that \sin x - \cos x = \frac{1}{\sqrt{5}}, what is the exact v… — JAMB UTME

If $x$ is an acute angle such that $\sin x - \cos x = \frac{1}{\sqrt{5}}$ , what is the exact value of $\tan x + \cot x$ ?

The exact value of

\tan x + \cot x

\frac{5}{2}

Squaring both sides of

\sin x - \cos x = \frac{1}{\sqrt{5}}

yields

1 - 2\sin x \cos x = \frac{1}{5}

, which simplifies to

\sin x \cos x = \frac{2}{5}

. Expressing

\tan x + \cot x

in terms of sine and cosine gives

\frac{\sin^2 x + \cos^2 x}{\sin x \cos x} = \frac{1}{\sin x \cos x}

. Substituting

\frac{2}{5}

gives

\frac{1}{2/5} = \frac{5}{2}

Square both sides of the given equation

\sin x - \cos x = \frac{1}{\sqrt{5}}

(\sin x - \cos x)^2 = \left(\frac{1}{\sqrt{5}}\right)^2 \implies \sin^2 x - 2\sin x \cos x + \cos^2 x = \frac{1}{5}

Squaring enables the use of the Pythagorean trigonometric identity.

Apply the fundamental identity

\sin^2 x + \cos^2 x = 1

to isolate

\sin x \cos x

1 - 2\sin x \cos x = \frac{1}{5} \implies 2\sin x \cos x = 1 - \frac{1}{5} = \frac{4}{5} \implies \sin x \cos x = \frac{2}{5}

This determines the value of the product of sine and cosine.

Rewrite

\tan x + \cot x

using quotient identities

\tan x + \cot x = \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} = \frac{\sin^2 x + \cos^2 x}{\sin x \cos x} = \frac{1}{\sin x \cos x}

Expressing tangent and cotangent with a common denominator simplifies the expression into a reciprocal.

Substitute

\sin x \cos x = \frac{2}{5}

into the simplified expression

\tan x + \cot x = \frac{1}{2/5} = \frac{5}{2}

Inverting the fraction gives the final numerical answer.

Pythagorean and Quotient Trigonometric Identities

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