NIMCET 2018 — Mathematics PYQ

Concept:
• tan θ = sin θ / cos θ.

Calculation:
Consider the relation in the set P = {θ: sin θ - cos θ = √2 cos θ}.
sin θ - cos θ = √2 cos θ
Dividing both sides by cos θ, we get:
tan θ - 1 = √2
⇒ tan θ = √2 + 1
∴ P = {θ: tan θ = √2 + 1} ... (1)
Consider the relation in the set Q = {θ: sin θ + cos θ = √2 sin θ}.

$$\sin \theta +\cos \theta =\sqrt{2}\sin \theta$$
Dividing both sides by cos θ, we get:
$$<br>\tan \theta +1=\sqrt{2}\tan \theta$$
$$\Rightarrow \tan \theta =\frac{1}{\sqrt{2}-1}=\frac{1}{\sqrt{2}-1}\times \frac{\sqrt{2}+1}{\sqrt{2}+1}=\frac{\sqrt{2}+1}{2-1}=\sqrt{2}+1$$
∴ Q = {θ: tan θ = √2 + 1}
$$<br>\therefore Q=\{\theta :\tan \theta =\sqrt{2}+1\}$$
... (2)
Comparing equations (1) and (2), we have:
P = Q.