Find x if, 2^sin²x + 2^cos²x = 3
"2^{\\sin^2x}+2^{\\cos^2x}=3"
"2^{\\sin^2x}+2^{1-\\sin^2x}=3"
"2^{\\sin^2x}+2\\cdot\\frac{1}{2^{\\sin^2x}}=3"
Let "y=2^{\\sin^2x}"
"y+\\frac{2}{y}=3"
"y^2-3y+2=0"
"y_1=1", "y_2=2"
"2^{\\sin^2x}=1"
"\\sin^2x=0"
"\\sin x=0"
"x=\\pi n, n\\in\\mathbb{Z}"
"2^{\\sin^2x}=2"
"\\sin^2x=1"
"\\sin x=\\pm1"
"x=(-1)^n\\frac{\\pi}{2}+\\pi n, n\\in\\mathbb{Z}"
Answer:
"x=\\pi n, n\\in\\mathbb{Z}"
"x=(-1)^n\\frac{\\pi}{2}+\\pi n, n\\in\\mathbb{Z}"
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