Question

Which statement is true for all real values of theta?

Answer 1

Answer:

  (a)  sec²(θ) -tan²(θ) = 1

Step-by-step explanation:

The identity relation between sec(θ) and tan(θ) is ...

  tan²(θ) +1 = sec²(θ)

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When tan²(θ) is subtracted from both sides of this equation, the result matches the first choice:

  sec²(θ) -tan²(θ) = 1

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Additional comment

This is a variation of the "Pythagorean" relationship between sine and cosine. There is a similar relation between cot²(θ) and csc²(θ).

  $\sin^2\theta+\cos^2\theta=1\\\\\dfrac{\sin^2\theta}{\cos^2\theta}+\dfrac{\cos^2\theta}{\cos^2\theta}=\dfrac{1}{\cos^2\theta}\qquad\text{divide by \cos^2\theta }\\\\\tan^2\theta+1=\sec^2\theta$