Use basic identities to simplify the expression. one

Ответ:

mzynique2000

31.12.2021

The simplified expression is:

$\dfrac{1}{\cot^2 \theta}+\sec \theta\cos \theta=\sec^2 \theta$

Step-by-step explanation:

We are asked to simplify the expression:

one divided by cotangent of theta to the second power+ sec θ cos θ

i.e. mathematically it is written as:

$=\dfrac{1}{\cot^2 \theta}+\sec \theta\cos \theta$

We know that:

$\tan \theta=\dfrac{1}{\cot \theta}\\\\i.e.\\\\(\tan \theta)^2=(\dfrac{1}{\cot \theta})^2\\\\i.e.\\\\\tan^2 \theta=\dfrac{1}{\cot^2 \theta}$

Hence, we can write this expression as:

$\dfrac{1}{\cot^2 \theta}+\sec \theta\cos \theta=\tan^2 \theta+\sec \theta\cos \theta$

Also, we know that:

$\sec \theta=\dfrac{1}{\cos \theta}$

Hence, we have:

$\sec \theta\cos \theta=\dfrac{\cos \theta}{\cos \theta}\\\\\\i.e.\\\\\\\sec \theta\cos \theta=1$

Hence, we get:

$=\dfrac{1}{\cot^2 \theta}+\sec \theta\cos \theta=\tan^2 \theta+1$

Also, we know that:

$\sec^2 \theta-\tan^2 \theta=1\\\\i.e.\\\\\sec^2 \theta=1+\tan^2 \theta$

Hence, we get the simplified expression as:

$\dfrac{1}{\cot^2 \theta}+\sec \theta\cos \theta=\sec^2 \theta$

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