Question

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

Answer 1

The simplified form of the expression is

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

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 = (\sin \theta)/(\cos\theta)`

Substitute in place of
`\tan\theta`

`=cos^2\theta(1+((\sin \theta)/(\cos\theta))^2)`

`=cos^2\theta(1+((\sin^2 \theta)/(\cos^2\theta)))`

Step 3 - Taking LCM,

`=cos^2\theta((\cos^2\theta+\sin^2 \theta)/(\cos^2\theta))`

`=cos^2\theta((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