Question

This one is of trigonometry please answer step by step with pic please answer

Answer 1

Sᴏʟᴜᴛɪᴏɴ ❶ :-

→ cos²θ(1 + tan²θ)

→ cos²θ + cos²θ*tan²θ

Using tan²A = (sin²A/cos²A) Now,

→ cos²θ + cos²θ*(sin²θ/cos²θ)

→ cos²θ + sin²θ

Now, we know that, sin²A + cos²A = 1

→ 1 = RHS (Proved).

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Sᴏʟᴜᴛɪᴏɴ ❷ :-

→ cos²θ(1 + tan²θ)

using (1 + tan²A) = sec²A we get,

→ cos²θ * sec²θ

Now, using sec²A = (1/cos²A) , we get,

→ cos²θ * (1/cos²θ)

→ 1 = RHS (Proved).

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Answer 2

$\star\;\sf{To\;Prove:}$

• $\sf{ { \cos}^2{ \theta}(1 + \tan^2{ \theta}) = 1}$

$\star\;\sf{Solution:}$

★ Taking L.H.S:-

$:\implies{\underline{\underline{\sf{\red{ { \cos}^2{ \theta}(1 + \tan^2{ \theta})}}}}}\\\\ :\implies\sf{ \cos^2{ \theta} \times \tan^2{ \theta}}$

✦ We know that,

${\underline{\boxed{\sf{ \tan^2{ \theta} = \dfrac{ \sin^2{ \theta}}{ \cos^2{ \theta}}}}}}$

Putting the value of $\sf{ \tan^2{ \theta}\; as\; \dfrac{ \sin^2{ \theta}}{ \cos^2{ \theta}}}$:

$:\implies{ \cos^2{ \theta} + \cancel{ \cos^2{ \theta}} \bigg( \dfrac{ \sin^2{ \theta}}{ \cancel{ \cos^2{ \theta}}} \bigg)}\\\\ :\implies\sf{ \cos^2{ \theta} + sin^2{ \theta}}$

✦ We know that,

$\;\;\;\star\;\sf{ \cos^2{ \theta} + sin^2{ \theta} = 1}$

✧ Therefore, L.H.S = R.H.S

${\underline{\underline{\sf{\purple{\dag\;Hence\;Proved!}}}}}$

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