Simplify the expression. tan 2 θ − sec 2 θ + 1 \tan^2\theta-\sec^2\theta+1 tan 2 θ − sec 2 θ + 1

In this problem, we're asked to simplify the expression. And the expression that we're given is the tangent squared of theta minus the secant squared of theta plus 1. Now remember that in order for a trig expression to be fully simplified, all of our arguments must be positive, which here they already are. Our expression should contain no fractions with as few trig functions as possible. Now currently, we don't have any fractions, so we're looking good so far, but let's see if we can do anything else. Remember that we want to be constantly scanning for identities. And looking up here, I see that I have 2 squared trig functions. And whenever I have squared trig functions I should be thinking about my pythagorean trig identities. Remember one of my pythagorean trig identities tells me that the tangent squared of theta plus 1 is equal to the secant squared of theta. Now we could use this in multiple forms, but let's come back up to our original expression here. Here I have the tangent squared of theta plus 1. So let's do some rearranging here. If I have the tangent squared of theta plus 1 minus the secant squared of theta, using that pythagorean trig identity, this tangent squared of theta plus 1 can just become the secant squared of theta. Then I just have the secant squared of theta minus the secant squared of theta. Now what if I take something and subtract that same something? Well, I'm just left with nothing. I'm left with 0 here. So I have no fractions, and I definitely have as few trig functions as possible. This is simply 0. Thanks for watching, and I'll see you in the next one.

Simplify the expression. tan ⁡ 2 θ − sec ⁡ 2 θ + 1 \tan^2\theta-\sec^2\theta+1 tan 2 θ − sec 2 θ + 1

csc ⁡ 2 θ + 1 \csc^2\theta+1 csc 2 θ + 1

6. Trigonometric Identities and More Equations

Introduction to Trigonometric Identities

Trigonometry

6. Trigonometric Identities and More Equations

Introduction to Trigonometric Identities

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Multiple Choice

Simplify the expression.
$$\tan$^2$\theta$-$\sec$^2$\theta$+1$

$$\csc$^2$\theta$+1$

Verified step by step guidance

Recall the Pythagorean identity: $ \sec^2\theta = 1 + \tan^2\theta $. This identity will help us simplify the expression.

Substitute $ \sec^2\theta $ in the expression $ \tan^2\theta - \sec^2\theta + 1 $ using the identity: $ \sec^2\theta = 1 + \tan^2\theta $.

The expression becomes $ \tan^2\theta - (1 + \tan^2\theta) + 1 $.

Simplify the expression by distributing the negative sign: $ \tan^2\theta - 1 - \tan^2\theta + 1 $.

Combine like terms: $ \tan^2\theta - \tan^2\theta + 1 - 1 $, which simplifies to 0.

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Simplify the expression.tan2θ−sec2θ+1\(\tan\)^2\(\theta\)-\(\sec\)^2\(\theta\)+1tan2θ−sec2θ+1

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Simplify the expression.
$\(\tan\)^2\(\theta\)-\(\sec\)^2\(\theta\)+1$