Trigonometric Functions And Graphs

Question

The fundamental period of the composite function $g(x) = \sin^4 x + \cos^4 x$, after simplification using power-reduction identities, is correctly given by which value?

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The key is to simplify the quartic combination into a single low-order cosine term, because the period of a sum of trigonometric powers is dictated by the highest multiple-angle that survives reduction, not by the original functions. Begin with the algebraic identity $\sin^4 x + \cos^4 x = (\sin^2 x + \cos^2 x)^2 - 2\sin^2 x\cos^2 x$, where the first bracket is one, leaving $1 - 2\sin^2 x\cos^2 x$. Recognizing $2\sin x\cos x = \sin 2x$, the term $2\sin^2 x\cos^2 x = \frac{1}{2}\sin^2 2x$, so the function is $1 - \frac{1}{2}\sin^2 2x$. Applying power reduction once more with $\sin^2 2x = \frac{1 - \cos 4x}{2}$ gives $1 - \frac{1}{4}(1 - \cos 4x) = \frac{3}{4} + \frac{1}{4}\cos 4x$. The periodicity is now governed entirely by the $\cos 4x$ term, whose fundamental period is $\frac{2\pi}{4} = \frac{\pi}{2}$. Option $\pi$ would correspond to a surviving $\cos 2x$ term, which does not appear after full reduction. Option $2\pi$ is the period of the unreduced sine and cosine. Option $\frac{\pi}{4}$ erroneously doubles the angular frequency. This follows the standard power-reduction periodicity pattern of JEE Advanced. As a final check, $g(0) = 1$ and $g\left(\frac{\pi}{2}\right) = 0 + 1 = 1$, and the function values repeat at spacing $\frac{\pi}{2}$, confirming the fundamental period.

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Given that $tan^{- 1} x + tan^{- 1} y = \frac{π}{4}$ holds for positive reals with $x y < 1$ , the e...

If $cos θ = \frac{3}{5}$ with $θ$ in the first quadrant, then applying the half-angle iden...

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