Identify the most helpful first step in verifying the identity.sec3θ=secθ+tan2θcosθ\sec^3\theta=\sec\theta+\frac{\tan^2\theta}{\cos\theta}sec3θ=secθ+cosθtan2θ

Use reciprocal identity to rewrite sec ⁡ θ \sec\theta sec θ on right side of equation.

Identify the most helpful first step in verifying the identity. sec 3 θ = sec θ + tan 2 θ cos θ \sec^3\theta=\sec\theta+\frac{\tan^2\theta}{\cos\theta} sec 3 θ = sec θ + c o s θ t a n 2 θ

In this problem, we're asked to identify the most helpful first step in verifying this identity. Now remember, just because it's the most helpful first step does not mean that it's the only first step that you could take. So if you had another idea for what you could do first, that's totally fine. You should go ahead and try to work through that in verifying this identity. Now looking at this identity that we have here, I have the secant cubed of theta is equal to the secant of theta plus the tangent squared of theta over the cosine of theta. So what should we do first? Well, remember when working with verifying identities, we wanna start with our more complicated side first. Now here, my more complicated side is clearly this right side because there are multiple terms and multiple trig functions. Now looking at this side, what should I do first? Well, see here I see that I have the secant of theta. Now remember that the secant of theta is just one over the cosine of theta. Now that would give this term the exact same denominator as the other term on this side. So this is going to be the most helpful first step in getting us started in verifying this identity because when I do that, when I replace that secant of theta with 1 over the cosine, I can then just combine these two fractions because they have a common denominator. Now that we've seen this, thanks for watching and I'll see you in the next one.

Identify the most helpful first step in verifying the identity. sec ⁡ 3 θ = sec ⁡ θ + tan ⁡ 2 θ cos ⁡ θ \sec^3\theta=\sec\theta+\frac{\tan^2\theta}{\cos\theta} sec 3 θ = sec θ + c o s θ t a n 2 θ

Use reciprocal identity to rewrite sec ⁡ θ \sec\theta sec θ on right side of equation.

Subtract sec ⁡ θ \sec\theta sec θ from both sides.

Rewrite tan ⁡ 2 θ \tan^2\theta tan 2 θ in terms of sine and cosine.

0. Functions

Trigonometric Identities

Calculus

0. Functions

Trigonometric Identities

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Multiple Choice

Identify the most helpful first step in verifying the identity.
$\sec^3\theta=\sec\theta+\frac{\tan^2\theta}{\cos\theta}$

Rewrite left side of equation in terms of sine and cosine.

Subtract $\sec\theta$ from both sides.

Use reciprocal identity to rewrite $\sec\theta$ on right side of equation.

Rewrite $\tan^2\theta$ in terms of sine and cosine.

Verified step by step guidance

Start by rewriting the left side of the equation $ \sec^3\theta $ in terms of sine and cosine. Recall that $ \sec\theta = \frac{1}{\cos\theta} $, so $ \sec^3\theta = \left(\frac{1}{\cos\theta}\right)^3 = \frac{1}{\cos^3\theta} $.

Next, rewrite the right side of the equation. Use the reciprocal identity to express $ \sec\theta $ as $ \frac{1}{\cos\theta} $.

Rewrite $ \tan^2\theta $ in terms of sine and cosine. Recall that $ \tan\theta = \frac{\sin\theta}{\cos\theta} $, so $ \tan^2\theta = \left(\frac{\sin\theta}{\cos\theta}\right)^2 = \frac{\sin^2\theta}{\cos^2\theta} $.

Substitute these expressions back into the equation. The right side becomes $ \frac{1}{\cos\theta} + \frac{\sin^2\theta}{\cos^3\theta} $.

Now, combine the terms on the right side over a common denominator. The common denominator is $ \cos^3\theta $, so the expression becomes $ \frac{\cos^2\theta + \sin^2\theta}{\cos^3\theta} $. Use the Pythagorean identity $ \cos^2\theta + \sin^2\theta = 1 $ to simplify the numerator.

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