Question

Solve the equation cos x - sin x = 1

Answer 1

Solution:
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$cos \: x \times \frac{1}{ \sqrt{2} } - \sin(x) \times \frac{1}{ \sqrt{2} } = \frac{1}{ \sqrt{2} } \\ \\ \cos(x) \cos(45°) - \sin(x) \sin(45°) = \frac{1}{ \sqrt{2} }$

on multiplying both side by 1/√2 .

now we are applying the formula of
cos a cos b - sin a sin b

$\cos(A) \cos(B) - \sin(A) \sin(B) = \cos(A + B)$

$\cos(x) \cos(45°) - \sin(x) \sin(45°) = \frac{1}{ \sqrt{2} } \\ \\ \cos(x + 45°) = \frac{1}{ \sqrt{2} } \\ \\ x + 45° = {cos}^{ - 1}( \frac{1}{ \sqrt{2} } ) \\ \\ x + 45° = {cos}^{ - 1}( \cos(45°) ) \\ \\ x + 45° = 45° \\ \\ x = 45° - 45° \\ \\ x = 0$