Question

OR
Prove that: (sin A + cosec A)2 + (COS A + sec A)2 = 7+ tan? A + cot? A.​

Answer 1

Step-by-step explanation:

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Answer 2

$\bf\huge\blue{\underline{\underline{ Question : }}}$

Prove that

(sin A + cosec A)² + (cos A + sec A)² = 7 + tan² A + cot² A

$\bf\huge\blue{\underline{\underline{ Solution : }}}$

★ Given that,

To prove that,

(sin A + cosec A)² + (cos A + sec A)² = 7 + tan² A + cot² A

★ Let,

LHS =

$\sf\:\implies (\sin\:A + \csc\:A)^{2} + (\cos\:A + \sec\:A)^{2}$

• (a + b)² = a² + b² + 2ab.

$\sf\:\implies (\sin^{2}\:A + \csc^{2}\:A + 2\sin\:A\csc\:A) + (\cos^{2}\:A + \sec^{2}\:A + 2\cos\:A\sec\:A)$

• csc A = 1/sin A
• sec A = 1/cos A

$\sf\:\implies (\sin^{2}\:A + \csc^{2}\:A + 2\times\sin\:A\times\cfrac{1}{\sin\:A})+(\cos^{2}\:A+\sec^{2}\:A + 2\times \cos\:A\times\cfrac{1}{\cos\:A})$

$\sf\:\implies \sin^{2}\:A+\csc^{2}\:A+2+\cos^{2}\:A+\sec^{2}\:A+2$

• sin² A + cos² A = 1

$\sf\:\implies \csc^{2}\:A+\sec^{2}\:A+1+2+2$

$\sf\:\implies \csc^{2}\:A+\sec^{2}\:A+5$

• csc² A = 1 + cot² A
• sec² A = 1 + tan² A

$\sf\:\implies 1 + \cot^{2}\:A+1+\tan^{2}\:A + 5$

$\sf\:\implies 7+ \cot^{2}\:A+\tan^{2}\:A$

Hence,it was proved.

★ More Information,

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

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