Prove the given identity: (1+ tan2 0) cos2 0=1 Which of the ‘ following shows the key – ‘steps in proving the identity? 0 A tan 2 0) cos ? 0 = cos 2 0 + tan 2 0cos 2 0 = cos 2 0 + sin 2 0=1 cos 2 0 0 B. (1+ tan2 0) cos2 0 = sec ? 0cos? 0 = cos 20=- csc 20 cos 2 0 (1+ tan2 0) cos? 0 = cot? 0cos 0 = cos 20 = cos tan 20 cos 2 0 cos 20 (1 – tan 2 0) cos 2 0 = sin20 + sin 20 = cos 2 0 + sin 2 0 =1 sin 20
The Correct Answer and Explanation is:
The correct answer is Option A, as it follows a structured approach to proving the given identity.
To prove the trigonometric identity:
(1+tan2θ)cos2θ=1(1 + \tan^2 \theta) \cos^2 \theta = 1
begin with the fundamental Pythagorean identity:
1+tan2θ=sec2θ1 + \tan^2 \theta = \sec^2 \theta
Substituting sec2θ\sec^2 \theta for 1+tan2θ1 + \tan^2 \theta results in:
sec2θcos2θ\sec^2 \theta \cos^2 \theta
Since secant and cosine are reciprocal functions, secθ=1cosθ\sec \theta = \frac{1}{\cos \theta}, leading to:
1cos2θcos2θ\frac{1}{\cos^2 \theta} \cos^2 \theta
Multiplying these expressions simplifies to:
11
Thus, the given equation holds true. Option A outlines the same logical steps by expressing (1+tan2θ)cos2θ(1 + \tan^2 \theta) \cos^2 \theta as cos2θ+tan2θcos2θ\cos^2 \theta + \tan^2 \theta \cos^2 \theta. Recognizing the identity tan2θ=sin2θcos2θ\tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta}, the equation further transforms into:
cos2θ+sin2θ=1\cos^2 \theta + \sin^2 \theta = 1
This result aligns with the Pythagorean identity, proving the original equation.
Understanding the proof process relies on recognizing reciprocal relationships among trigonometric functions and their identity-based transformations. The equation elegantly demonstrates the consistency of fundamental trigonometric principles, reinforcing the interconnected nature of sine, cosine, tangent, and secant functions. Expressing the identity in multiple equivalent forms enhances problem-solving efficiency and confirms the validity of trigonometric properties across varying approaches.
