Solving $\sec\theta - 1 = \left(\sqrt{2} - 1\right) \tan\theta$

Question

Solve $$\sec\theta - 1 = \left(\sqrt{2} - 1\right) \tan\theta$$

My try :

Is there any process possible for this equation? Please share it.

Answer 1

Hint:

$$\dfrac{\sec2y-1}{\tan2y}=\dfrac{1-\cos2y}{\sin2y}=\tan y$$

Alternatively, $$\sqrt2-1=\dfrac{\sec2y-1}{\tan2y}=\csc2y-\cot2y$$

$$\iff\csc2y+\cot2y=\dfrac1{\csc2y-\cot2y}=?$$

Answer 2

HINT:

Dividing tan theta both side you get - $$\frac{1-\cos{x}}{\sin{x}}= \sqrt{2}-1$$ Now square it on both sides , and solving further you will reach the solution.

Additional hints: $$\frac{1-\cos{x}}{1+\cos{x}}= 3-2\sqrt{2}$$ $$\cos{x} = \frac{1}{\sqrt{2}} $$ and you are done.