Trigonometric Identity Simplification
We are given the equation $\cos^4\theta - \sin^4\theta = \frac{2}{3}$ and asked to find the value of $1 - 2\sin^2\theta$.
Solving the Trigonometric Equation
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Begin with the provided equation:
$ \cos^4\theta - \sin^4\theta = \frac{2}{3} $
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Factor the expression on the left using the difference of squares formula ($a^4 - b^4 = (a^2 - b^2)(a^2 + b^2)$):
$ (\cos^2\theta - \sin^2\theta)(\cos^2\theta + \sin^2\theta) = \frac{2}{3} $
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Apply the fundamental Pythagorean identity, $\cos^2\theta + \sin^2\theta = 1$:
$ (\cos^2\theta - \sin^2\theta)(1) = \frac{2}{3} $
$ \cos^2\theta - \sin^2\theta = \frac{2}{3} $
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Recall the double angle identity for cosine: $\cos(2\theta) = \cos^2\theta - \sin^2\theta$. Another form of this identity is $\cos(2\theta) = 1 - 2\sin^2\theta$.
Therefore, the expression $\cos^2\theta - \sin^2\theta$ is equivalent to $1 - 2\sin^2\theta$.
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Using the result from step 3 and the identity from step 4, we can directly find the required value:
$ 1 - 2\sin^2\theta = \frac{2}{3} $
The value of $1 - 2\sin^2\theta$ is $\frac{2}{3}$.
Similar Questions
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Important Questions from Trigonometric Ratios and Identities
What is the ratio of the greatest to the smallest value of 2 – 2 sin x – sin 2x, 0 ≤ x ≤ (π/2)?
If sinθ = \(\frac{4}{5}\) , Find the value of sin3θ
If x, y are acute angles, where 0 < x + y < 90° and sin(3x - 40°) = cos (3y + 40°), then the value of tan (x + y) is equal to
What is (1 + cot θ - cosec θ)(1 + tan θ + sec θ) equal to?
If \(\sin \theta =\frac{3}{5}\) and \(\cos \theta =\frac{4}{5}\) , then the value of \(\frac{1+\tan \theta}{1-\cot \theta}\) is:
