Given that \tan \theta + \cot \theta = 4 for an acute angle \theta, what is the exact deci… — JAMB UTME

Given that $\tan \theta + \cot \theta = 4$ for an acute angle $\theta$ , what is the exact decimal value of $\sin^4 \theta + \cos^4 \theta$ ?

Answer: 0.875

The exact decimal value of

\sin^4 \theta + \cos^4 \theta

is 0.875.

By writing

\tan \theta + \cot \theta

\frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} = \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} = 4

, we find that

\sin \theta \cos \theta = \frac{1}{4}

. Squaring the fundamental identity

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

gives

\sin^4 \theta + 2\sin^2 \theta \cos^2 \theta + \cos^4 \theta = 1

. Isolating

\sin^4 \theta + \cos^4 \theta

yields

1 - 2(\sin \theta \cos \theta)^2 = 1 - 2\left(\frac{1}{16}\right) = 1 - \frac{1}{8} = \frac{7}{8} = 0.875

Rewrite the expression

\tan \theta + \cot \theta = 4

using sine and cosine ratios

\frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} = 4

Applies the fundamental quotient identities for tangent and cotangent.

Combine fractions over a common denominator and apply the Pythagorean identity

\frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} = \frac{1}{\sin \theta \cos \theta} = 4 \implies \sin \theta \cos \theta = \frac{1}{4}

Utilizes the identity

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

Relate

\sin^4 \theta + \cos^4 \theta

(\sin^2 \theta + \cos^2 \theta)^2

\sin^4 \theta + \cos^4 \theta = (\sin^2 \theta + \cos^2 \theta)^2 - 2\sin^2 \theta \cos^2 \theta = 1 - 2(\sin \theta \cos \theta)^2

Uses the algebraic identity

a^2 + b^2 = (a+b)^2 - 2ab

where

a = \sin^2 \theta

and

b = \cos^2 \theta

Substitute

\sin \theta \cos \theta = \frac{1}{4}

into the algebraic relation

1 - 2\left(\frac{1}{4}\right)^2 = 1 - 2\left(\frac{1}{16}\right) = 1 - \frac{1}{8} = \frac{7}{8} = 0.875

Evaluates the expression to obtain the final decimal result.

Trigonometric Identities and Algebraic Polynomial Expansion

Estimated Time:2m 0s

Question