Question

Q.2. Prove the following:-
( sec tetha + tan theta ) (1-sin theta) = cos theta​

Answer 1

Given Question :-

$\tt \:  Prove \: that \: (sec \theta + tan \theta)(1 - sin \theta) = cos \theta$

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Identities used :-

$\tt \:  ⟼1. \: sec \theta = \dfrac{1}{cos \theta}$

$\tt \:  ⟼2. \: tan \theta = \dfrac{sin \theta}{cos \theta}$

$\tt \:  ⟼3. \: 1 - {sin}^{2} \theta = {cos}^{2} \theta$

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☆ Solution:-

Consider LHS:-

$\tt \:  ⟼\: (sec \theta + tan \theta)(1 - sin \theta)$

$\tt \:  ⟼ \: = \: \bigg(\dfrac{1}{cos \theta} + \dfrac{sin \theta}{cos \theta} \bigg )(1 - sin \theta)$

$\tt \:  ⟼ \: = \: \bigg(\dfrac{1 + sin \theta}{cos \theta} \bigg) (1 - sin \theta)$

$\tt \:  ⟼ = \: \dfrac{1 - {sin}^{2} \theta }{cos \theta}$

$\tt \:  ⟼ = \: \dfrac{{cos}^{2} \theta }{cos \theta}$

$\tt \:  ⟼ = \: cos \theta$

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Additional Information:-

Relationship between sides and T ratios

sin θ = Opposite Side/Hypotenuse

cos θ = Adjacent Side/Hypotenuse

tan θ = Opposite Side/Adjacent Side

sec θ = Hypotenuse/Adjacent Side

cosec θ = Hypotenuse/Opposite Side

cot θ = Adjacent Side/Opposite Side

Reciprocal Identities

cosec θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ

sin θ = 1/cosec θ

cos θ = 1/sec θ

tan θ = 1/cot θ

Co-function Identities

sin (90°−x) = cos x

cos (90°−x) = sin x

tan (90°−x) = cot x

cot (90°−x) = tan x

sec (90°−x) = cosec x

cosec (90°−x) = sec x

Fundamental Trigonometric Identities

sin²θ + cos²θ = 1

sec²θ - tan²θ = 1

cosec²θ - cot²θ = 1