Question

(10.05 MC) Simplify cos2 Θ(1 + tan2 Θ). (2 points)

Answer 1

Answer:

The simplified form of the expression is

$\cos^2\theta(1+\tan^2\theta)=1$

Step-by-step explanation:

Given : Expression $\cos^2\theta(1+\tan^2\theta)$

To find : The simplification of the expression?

Solution :

Step 1 - Write the expression

$cos^2\theta(1+\tan^2\theta)$

Step 2 - Using the trigonometric property,

$\tan\theta = \frac{\sin \theta}{\cos\theta}$

Substitute in place of $\tan\theta$

$=cos^2\theta(1+(\frac{\sin \theta}{\cos\theta})^2)$

$=cos^2\theta(1+(\frac{\sin^2 \theta}{\cos^2\theta}))$

Step 3 - Taking LCM,

$=cos^2\theta(\frac{\cos^2\theta+\sin^2 \theta}{\cos^2\theta})$

$=cos^2\theta(\frac{1}{\cos^2\theta})$

Step 4 - Cancel the term

$=1$

Therefore, The simplified form of the expression is

$\cos^2\theta(1+\tan^2\theta)=1$

Answer 2

Since tanΘ = sinΘ / cosΘ:
cos^2 Θ (1 + tan^2 Θ)
= cos^2 Θ (1 + sin^2 Θ / cos^2 Θ)
Distributing:
= cos^2 Θ + sin^2 Θ
By the Pythagorean identity,
= 1