If y = x + tan x. Show that cos2x . \(\frac{dy}{dx}\) = 2 – sin2x

1.	If y = x + tan x. Show that cos2x . \(\frac{dy}{dx}\) = 2 – sin2x
Answer» Given y = x + tan x \(\frac{dy}{dx}\) = 1 + sec²x = 1 + \(\frac{1}{cos^2x}\) = \(\frac{cos^2x + 1}{cos^2x}\) ∴ cos²x\(\frac{dy}{dx}\) = cos²x + 1 = 1 – sin²x + 1 = 2 – sin²x ∴ cos²x \(\frac{dy}{dx}\) = 2 – sin²x

Answer»

Given y = x + tan x

\(\frac{dy}{dx}\) = 1 + sec²x = 1 + \(\frac{1}{cos^2x}\)

= \(\frac{cos^2x + 1}{cos^2x}\)

∴ cos²x\(\frac{dy}{dx}\) = cos²x + 1 = 1 – sin²x + 1

= 2 – sin²x

∴ cos²x \(\frac{dy}{dx}\) = 2 – sin²x

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