Number of solutions of the equation $$(3+1)2x+(3$$−$$1)2x=23x_____$$.

$$(3+1)2x+(3$$−$$1)2x=23x=(22)2x$$⇒$$3+1222x+3$$−$$1222x=1$$⇒$$sin$$⁡$$75$$∘$$2x+$$$$cos$$⁡$$75$$∘$$2x=1$$⇒$$x=1$$

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Number of solutions of the equation $(\sqrt{3} + 1)^{2 x} + (\sqrt{3} - 1)^{2 x} = 2^{3 x}$ _____.

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Solution

$\begin{array}{l} (\sqrt{3} + 1)^{2 x} + (\sqrt{3} - 1)^{2 x} = 2^{3 x} = (2 \sqrt{2})^{2 x} \\ \Rightarrow {(\frac{\sqrt{3} + 1}{2 \sqrt{2}})}^{2 x} + {(\frac{\sqrt{3} - 1}{2 \sqrt{2}})}^{2 x} = 1 \\ \Rightarrow {(\sin 75^{\circ})}^{2 x} + {(\cos 75^{\circ})}^{2 x} = 1 \Rightarrow x = 1 \end{array}$

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Compound angles

Remember concepts with our Masterclasses.

Number of solutions of the equation (3+1)2x+(3−1)2x=23x_____.

(3+1)2x+(3−1)2x=23x=(22)2x⇒3+1222x+3−1222x=1⇒sin⁡75∘2x+cos⁡75∘2x=1⇒x=1

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Number of solutions of the equation $(\sqrt{3} + 1)^{2 x} + (\sqrt{3} - 1)^{2 x} = 2^{3 x}$ _____.