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Exercise

$cosx\left(2secx+tanx\right)\left(secx-2tanx\right)$

Step-by-step Solution

1

Multiply the single term $\cos\left(x\right)\left(\sec\left(x\right)-2\tan\left(x\right)\right)$ by each term of the polynomial $\left(2\sec\left(x\right)+\tan\left(x\right)\right)$

$2\sec\left(x\right)\cos\left(x\right)\left(\sec\left(x\right)-2\tan\left(x\right)\right)+\tan\left(x\right)\cos\left(x\right)\left(\sec\left(x\right)-2\tan\left(x\right)\right)$
2

Multiply the single term $2\sec\left(x\right)\cos\left(x\right)$ by each term of the polynomial $\left(\sec\left(x\right)-2\tan\left(x\right)\right)$

$2\sec\left(x\right)^2\cos\left(x\right)-4\tan\left(x\right)\sec\left(x\right)\cos\left(x\right)+\tan\left(x\right)\cos\left(x\right)\left(\sec\left(x\right)-2\tan\left(x\right)\right)$
3

Multiply the single term $\tan\left(x\right)\cos\left(x\right)$ by each term of the polynomial $\left(\sec\left(x\right)-2\tan\left(x\right)\right)$

$2\sec\left(x\right)^2\cos\left(x\right)-4\tan\left(x\right)\sec\left(x\right)\cos\left(x\right)+\sec\left(x\right)\tan\left(x\right)\cos\left(x\right)-2\tan\left(x\right)^2\cos\left(x\right)$
4

Combining like terms $-4\tan\left(x\right)\sec\left(x\right)\cos\left(x\right)$ and $\sec\left(x\right)\tan\left(x\right)\cos\left(x\right)$

$2\sec\left(x\right)^2\cos\left(x\right)-3\sec\left(x\right)\tan\left(x\right)\cos\left(x\right)-2\tan\left(x\right)^2\cos\left(x\right)$
5

Applying the trigonometric identity: $\cos\left(\theta \right)\sec\left(\theta \right) = 1$

$2\sec\left(x\right)^2\cos\left(x\right)-3\tan\left(x\right)-2\tan\left(x\right)^2\cos\left(x\right)$

Final answer to the exercise

$2\sec\left(x\right)^2\cos\left(x\right)-3\tan\left(x\right)-2\tan\left(x\right)^2\cos\left(x\right)$

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