If $\cos^4 heta - \sin^4 heta = k $, then the value of $\frac{1-k}{1+k}$ is:

Question

If $\cos^4 	heta - \sin^4 	heta = k $, then the value of $\frac{1-k}{1+k}$ is:

Examoogle · Accepted Answer

The correct answer is Option A: $	an^2 	heta$. Simplifying Expression (1 - k)/(1 + k) when cos⁴θ - sin⁴θ = k

Step-by-Step Solution & Concepts:
Key Fact: Simplifying Expression (1 - k)/(1 + k) when cos⁴θ - sin⁴θ = k

Given: $\cos^4 	heta - \sin^4 	heta = k$

Key Formulas: $\cos^4 	heta - \sin^4 	heta = \cos 2	heta$; $\frac{1 - \cos 2	heta}{1 + \cos 2	heta} = 	an^2 	heta$

Key Points: Alternative check: plug in $	heta = 45^\circ$ $\implies k = 0 \implies (1-0)/(1+0) = 1 = 	an^2 45^\circ$.

Steps: Factorize Given Equation: $$\cos^4 	heta - \sin^4 	heta = (\cos^2 	heta - \sin^2 	heta)(\cos^2 	heta + \sin^2 	heta)$$ Since $\cos^2 	heta + \sin^2 	heta = 1$ and $\cos^2 	heta - \sin^2 	heta = \cos 2	heta$: $$k = \cos 2	heta$$. Evaluate (1 - k) / (1 + k): $$\frac{1 - k}{1 + k} = \frac{1 - \cos 2	heta}{1 + \cos 2	heta}$$ Using standard half-angle identities $1 - \cos 2	heta = 2\sin^2 	heta$ and $1 + \cos 2	heta = 2\cos^2 	heta$: $$\frac{1 - k}{1 + k} = \frac{2\sin^2 	heta}{2\cos^2 	heta} = 	an^2 	heta$$

Answer: Option A is correct — tan²θ

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