Prove the identity 4 theta + 2 theta = 4 theta - 2 theta.

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ScanSolve AI Tutor · Accepted Answer

To prove the identity tan⁴ θ + tan² θ = sec⁴ θ − sec² θ. Step 1: Start with the Left Hand Side (LHS) of the equation. LHS = tan⁴ θ + tan² θ Step 2: Factor out tan² θ from the expression. LHS = tan² θ (tan² θ + 1); Step 3: Use the trigonometric identity 1 + tan² θ = sec² θ. Substitute this into the expression. LHS = tan² θ (sec² θ); Step 4: Use the same identity to express tan² θ in terms of sec² θ, which is tan² θ = sec² θ − 1. Substitute this into the expression. LHS = (sec² θ − 1) (sec² θ) Step 5: Distribute sec² θ across the terms in the parenthesis. LHS = sec² θ · sec² θ − 1 · sec² θ; LHS = sec⁴ θ − sec² θ Step 6: Compare the result with the Right Hand Side (RHS) of the original equation. RHS = sec⁴ θ − sec² θ; Since LHS = RHS, the identity is proven. tan⁴ θ + tan² θ = sec⁴ θ − sec² θ

Prove the identity 4 theta + 2 theta = 4 theta - 2 theta.

Step-by-step solution

Start with the Left Hand Side (LHS) of the equation.

Factor out $\tan^2 \theta$ from the expression.

Use the trigonometric identity $1 + \tan^2 \theta = \sec^2 \theta$ .

Use the same identity to express $\tan^2 \theta$ in terms of $\sec^2 \theta$ , which is $\tan^2 \theta = \sec^2 \theta - 1$ .

Distribute $\sec^2 \theta$ across the terms in the parenthesis.

Compare the result with the Right Hand Side (RHS) of the original equation.

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More Mathematics Questions

Prove the identity 4 theta + 2 theta = 4 theta - 2 theta.

Step-by-step solution

Start with the Left Hand Side (LHS) of the equation.

Factor out tan⁡2θ\tan^2 \thetatan2θ from the expression.

Use the trigonometric identity 1+tan⁡2θ=sec⁡2θ1 + \tan^2 \theta = \sec^2 \theta1+tan2θ=sec2θ.

Use the same identity to express tan⁡2θ\tan^2 \thetatan2θ in terms of sec⁡2θ\sec^2 \thetasec2θ, which is tan⁡2θ=sec⁡2θ−1\tan^2 \theta = \sec^2 \theta - 1tan2θ=sec2θ−1.

Distribute sec⁡2θ\sec^2 \thetasec2θ across the terms in the parenthesis.

Compare the result with the Right Hand Side (RHS) of the original equation.

Still stuck on this one?

More Mathematics Questions

Factor out $\tan^2 \theta$ from the expression.

Use the trigonometric identity $1 + \tan^2 \theta = \sec^2 \theta$ .

Use the same identity to express $\tan^2 \theta$ in terms of $\sec^2 \theta$ , which is $\tan^2 \theta = \sec^2 \theta - 1$ .

Distribute $\sec^2 \theta$ across the terms in the parenthesis.