Number of values of x∈$$0$$,$$4$$π and satisfying $$2secx+tanx=1$$ is

We can write the given equation as $$2=cosx$$−sinx⇒$$12cosx$$−$$12sinx=1$$⇒$$cosx+$$π$$4=1$$⇒$$x+$$π$$4=2n$$π, n∈I⇒$$x=7$$$$π$$$$4$$,$$15$$$$π$$$$4$$

Trigonometric equations

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Question

Number of values of $x \in [0, 4 π]$ and satisfying $\sqrt{2} \sec x + \tan x = 1$ is

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Solution

We can write the given equation as $\sqrt{2} = \cos x - \sin x$
$\Rightarrow \frac{1}{\sqrt{2}} \cos x - \frac{1}{\sqrt{2}} \sin x = 1$
$\Rightarrow \cos (x + \frac{π}{4}) = 1$
$\Rightarrow x + \frac{π}{4} = 2 n π, n \in I$
$\Rightarrow x = \frac{7 π}{4}, \frac{15 π}{4}$

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