Question

1)

Answer 1

$\qquad$ $\purple{\twoheadrightarrow\bf 2cos^2A = 1 + cos2A }$

$\qquad$ $\purple{\twoheadrightarrow\bf a^3 +b^3 = (a+b)^3 -3ab(a+b)}$

$\qquad$ $\purple{\twoheadrightarrow\bf 2 sinA cosA = sin2A}$

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1)

$\qquad$ $\twoheadrightarrow\bf L.H.S$

$\qquad$ $\pink{\twoheadrightarrow\bf cos^4 x}$

$\qquad$ $\twoheadrightarrow\sf \dfrac{1}{4} (2cos^2x)^2$

$\qquad$ $\twoheadrightarrow\sf \dfrac{1}{4}(1+cos2x)^2$

$\qquad$ $\twoheadrightarrow\sf \dfrac{1}{4}(1+2cos2x+cos^22x)$

$\qquad$ $\twoheadrightarrow\sf \dfrac{1}{4}+\dfrac{1}{2}cos2x+\dfrac{1}{8}\times 2cos^22x$

$\qquad$ $\twoheadrightarrow\sf \dfrac{1}{4}+\dfrac{1}{2}cos2x+\dfrac{1}{8}\times(1+cos4x)$

$\qquad$ $\pink{\twoheadrightarrow\bf \dfrac{1}{4}+\dfrac{1}{2}cos2x+\dfrac{1}{8}\times cos4x}$

$\qquad$ $\twoheadrightarrow\bf R.H.S$

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2)

$\qquad$ $\twoheadrightarrow\bf L.H.S$

$\qquad$ $\pink{\twoheadrightarrow\bf cos^6\theta +sin^6\theta }$

$\qquad$ $\twoheadrightarrow\sf (cos^2\theta)^3 +(sin^2\theta)^3$

$\twoheadrightarrow\sf (cos^2\theta +sin^2\theta)^3 -3cos^2\theta sin^2\theta (cos^2\theta +sin^2\theta)$

$\qquad$ $\twoheadrightarrow\sf 1-3\times \dfrac{1}{4}\times 4(sin\theta cos\theta) ^2$

$\qquad$ $\twoheadrightarrow\sf 1-\dfrac{3}{4}(2sin\theta cos\theta) ^2$

$\qquad$ $\pink{\twoheadrightarrow\sf 1-\dfrac{3}{4}sin^22\theta}$

$\qquad$ $\twoheadrightarrow\bf R.H.S$