Question

(Sin A+ sec A)^2+( cos A+ cosec A)^2= (1+sec A× cosec A)^2

Answer 1

${\boxed{(sinA+secA)^{2}+(cosA+cosecA)^{2}}}$

= sin²A + 2sinA × secA + sec²A + cos²A + 2cosAcosecA + cosec²A

$\boxed{Arrange\:in\:Sequence}$

= (sin²A + cos²A) + sec²A + cosec²A + 2tanA + 2cotA

$\boxed{Identity:sin^{2}A+cos^{2}A =1}$

= $\tt{\rightarrow 1+sec^{2}A + cosec^{2}A + 2[ \dfrac{sinA}{cosA}+\dfrac{cosA}{sinA}]}$

= 1 + sec²A + cosec²A + 2×secAcosecA

$\tt\boxed{{Identity:sec^{2}A+cosec^{2}A=sec^{2}A\times cosec^{2}A}}$

= (1 + secA × cosecA)²

$\textsc{Hence we get LHS = RHS}$

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★Some additional identities :-

$\tt{\rightarrow tan\theta =\dfrac{sin\theta}{cos\theta}}$

$\tt{\rightarrow cot\theta =\dfrac{cos\theta}{sin\theta}}$

★(sin²θ) + (cos² θ) = 1

★1 + tan² θ = sec² θ

★1 + cot² θ = cosec² θ