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
- If $\cot A = \sqrt{5}$, then find the value of $\frac{\csc^2 A - \sec^2 A}{\csc^2 A + \sec^2 A}$
- Find the value of $\frac{\cos^6 \theta + \sin^6 \theta + 3 \sin^2 \theta \cos^2 \theta}{\csc \theta \sec \theta ((\sin \theta + \cos \theta)^2 - 1)}$
- The value of $2\sin30^{\circ} + 5\cos60^{\circ} + 3\sin180^{\circ}$ is:
If $\tan\theta = \frac{5}{12}$, $0 < \theta < \frac{\pi}{2}$, then the value of $\frac{\cos\theta + 5\cot\theta}{\text{cosec}\theta - \cos\theta}$ will be:
- If $\sin(3A - 20^\circ) = \cos(20^\circ - 3B)$, then the value of A - B is:
- If $\tan A = \frac{8}{15}$ then find $\sin A + \cos A$.
- If $\sin \theta - \cos \theta = 0$, and $0 \le \theta \le 90^\circ$, then the value of $\sin \theta + \cos \theta$ is:
- If $1 + \tan \theta = \sqrt{3}$, then $\sqrt{3} \cot \theta - 1 = ?$
- $\cos(x - y) = \frac{\sqrt{3}}{2}$, and $\sin(x + y) = 1$, where $x$ and $y$ are positive acute angles and $x \ge y$, then $x$ and $y$ are:
- Simplify: $\frac{\sin A \tan A}{(1 - \cos A)}$
Important Questions from Trigonometric Ratios and Identities
If cosec θ = 13/12, then sin θ + cos θ - tan θ is equal to:
What is the value of \(\frac{3 \sin 58^{\circ}}{\cos 32^{\circ}}+\frac{3 \sin 42^{\circ}}{\cos 48^{\circ}}\) ?
If \(\frac{{\tan \theta + \sin \theta }}{{\tan \theta - \sin \theta }} = \frac{{k + 1}}{{k - 1}},\) then k = ?
If α + β = 90° and α = 2β, then the value of 3 cos 2 α - 2 sin 2 β is equal to:
If \(\sqrt{3}\) tan θ = 3 sin θ, then what is the value of sin 2θ − cos 2θ ?
