Question

If 4 cosx²+4 sin x = 5, show that sinx = 1/2​

Answer 1

See proof below

Explanation:

`4cos^2(x) +4sin(x) = 5\\\\\sf Divide\; both \; sides\; by\; 4\\== > cos^2(x) + sin(x) = (5)/(4) .................(1)\\\\cos^2(x) + sin^2(x) = 1\\\\cos^2(x) = 1- sin^2(x) \\\\\textsf {Substituting into equation (1) }\\1-sin^2(x) + sin(x) = (5)/(4)\\\\\\sf Subtract\; 1\; from\; both \; sides\\\mathsf{-\sin^2(x) + \sin(x) = (5)/(4) - 1}\\\\ (5)/(4) - 1 = (5)/(4) - (4)/(4) = (1)/(4)\\`

`\sf {-\sin^2(x) + \sin(x) = (1)/(4)\\`

`\sf Subtract\; (1)/(4)\;from\;both\;sides\\`

`{-\sin^2(x) + \sin(x) - (1)/(4) = 0\\\\`

Multiply throughout by -1

`\sf \sin^2(x) - \sin(x) + (1)/(4) = 0\\\\`

Let u = sin(x). Substituting we get

`\sf u^2 - u + (1)/(4) = 0`

This is a quadratic equation which can be solved using sum of squars

Note that

`\sf (u - (1)/(2))^2 = u^2 -2\cdot (1)/(4)u + ((1)/(2))^2 \\= \sf u^2 - u + (1)/(4) = 0`

So

`\sf (u - (1)/(2))^2 = 0\\Solution\; is \;u = (1)/(2)\\\\Substituting\sin(x) \;for\;u \;gives\\sin(x) = (1)/(2)`

PROVED