Question

Verify that (cos²a) (2 + tan² a) = 2 - sin² a....
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Answer 1

See below ~

Explanation:

Identities used :

⇒ cos²a = 1 - sin²a

⇒ tan²a = sin²a / cos²a

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Solving :

⇒ (cos²a) (2 + tan² a)

⇒ 2cos²a + (cos²a)(tan²a)

⇒ 2(1 - sin²a) + sin²a

⇒ 2 - 2sin²a + sin²a

⇒ 2 - sin²a [∴ Proved √]

Answer 2

Trigonometric Formula's:

`\boxed{\sf \ \sf \sin^2 \theta + \cos^2 \theta = 1}`

`\boxed{ \sf tan\theta = (sin\theta)/(cos\theta) }`

Given to verify the following:

`\bf (cos^2a) (2 + tan^2 a) = 2 - sin^2 a`

`\texttt{\underline{rewrite the equation}:}`

`\rightarrow \sf (cos^2a) (2 + (sin^2 a)/(cos^2 a) )`

`\texttt{\underline{apply distributive method}:}`

`\rightarrow \sf 2 (cos^2a) + ((sin^2 a)/(cos^2 a) ) (cos^2a)`

`\texttt{\underline{simplify the following}:}`

`\rightarrow \sf 2cos^2 a + sin^2 a`

`\texttt{\underline{rewrite the equation}:}`

`\rightarrow \sf 2(1 - sin^2a ) + sin^2 a`

`\texttt{\underline{distribute inside the parenthesis}:}`

`\rightarrow \sf 2 - 2sin^2a + sin^2 a`

`\texttt{\underline{simplify the following}} :`

`\rightarrow \sf 2 - sin^2a`

Hence, verified the trigonometric identity.