Question

(secA + tanA) (1-sinA) =cos A
prove​

Answer 1

(sec A + tan A)(1 - sin A) = cos A

Explanation:

★ Given that,

Prove that → (sec A + tan A)(1 - sin A) = cos A.

★ LHS :

$\sf \implies (\sec\:A + \tan\:A)(1-\sin\:A)$

• sec A = 1/cos A
• tan A = sin A/cos A

$\sf \implies \bigg ( \cfrac{1}{\cos\:A} + \cfrac{\sin\:A}{\cos\:A} \bigg )(1-\sin\:A)$

$\sf \implies \bigg ( \cfrac{1 + \sin\:A}{\cos\:A} \bigg)(1-\sin\:A)$

$\sf \implies \bigg ( \cfrac{(1 + \sin\:A)(1-\sin\:A)}{\cos\:A} \bigg )$

• (a + b)(a - b) = a² - b²

$\sf \implies \bigg ( \cfrac{(1)^{2} - (\sin\:A)^{2}}{\cos\:A} \bigg )$

$\sf \implies \bigg ( \cfrac{1 - \sin^{2}\:A}{\cos\:A} \bigg )$

• 1 - sin² A = cos² A

$\sf \implies \bigg( \cfrac{\cancel{cos^{2}\:A}}{\cancel{\cos\:A}} \bigg)$

$\sf \implies \cos\:A \tt\green{ \: \: = RHS}$

☯ Since, LHS = RHS.

∴ Hence, it was proved.

★ More Information :

$\boxed{\begin{minipage}{7 cm} Trigonometric Identities : \\ \\ \sin^{2}\theta + cos^{2}\theta = 1 \\ \\ 1 + tan^{2}\theta = sec^{2}\theta \\ \\1 + cot^{2}\theta=\text{cosec}^2\, \theta \end{minipage}}$

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