Question

Trigeonometric Identities
Exercise 13 A
Question number 31
( 1 + tan∅ + cot∅ ) ( sin ∅ - cos ∅ ) = ( sec∅ / cosec²∅ - cosec∅ / sec²∅ )

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Answer 1

To Proof:-

$\sf{(1 + tan\theta + cot\theta)(sin\theta - cos\theta) = \bigg(\dfrac{sec\theta}{cosec^2\theta} - \dfrac{cosec\theta}{sec^2\theta}\bigg)}$

Solution:-

Taking LHS:-

(1 + tanθ + cotθ)(sinθ - cosθ)

We know:-

$\dag{\boxed{\pink{\sf{tan\theta = \dfrac{sin\theta}{cos\theta}}}}}$

$\dag{\boxed{\pink{\sf{cot\theta = \dfrac{cos\theta}{sin\theta}}}}}$

Hence,

$\sf{\bigg(1 + \dfrac{sin\theta}{cos\theta} + \dfrac{cos\theta}{sin\theta}\bigg)(sin\theta - cos\theta)}$

$= \sf{(sin\theta - cos\theta)\bigg(1 + \dfrac{sin\theta}{cos\theta} + \dfrac{cos\theta}{sin\theta}\bigg)}$

$= \sf{sin\theta\bigg(1 + \dfrac{sin\theta}{cos\theta} + \dfrac{cos\theta}{sin\theta}\bigg) - cos\theta\bigg(1 + \dfrac{sin\theta}{cos\theta} + \dfrac{cos\theta}{sin\theta}\bigg)}$

$= \sf{ \dfrac{1}{cosec\theta} \bigg(1 + \dfrac{ \dfrac{1}{cosec \theta}}{ \dfrac{1}{sec \theta}} + \dfrac{ \dfrac{1}{sec \theta}}{ \dfrac{1}{cosec \theta}} \bigg) - \dfrac{1}{sec \theta} \bigg(1 + \dfrac{ \dfrac{1}{cosec \theta}}{ \dfrac{1}{sec \theta}} + \dfrac{ \dfrac{1}{cosec \theta}}{ \dfrac{1}{sec \theta}} \bigg)}$

$= \sf{\dfrac{1}{cosec\theta}\bigg(1 + \dfrac{sec\theta}{cosec\theta} + \dfrac{cosec\theta}{sec\theta}\bigg) - \dfrac{1}{sec\theta}\bigg(1 + \dfrac{sec\theta}{cosec\theta} + \dfrac{cosec\theta}{sec\theta}\bigg)}$

$=\sf{ \bigg[\dfrac{1}{cosec\theta} + \dfrac{sec\theta}{cosec^2\theta} + \dfrac{cosec\theta}{sec\theta} \times \dfrac{1}{cosec\theta}\bigg] - \bigg[\dfrac{1}{sec\theta} + \dfrac{sec\theta}{cosec\theta} \times \dfrac{1}{sec\theta} + \dfrac{cosec\theta}{sec^2\theta}\bigg]}$

$= \sf{\dfrac{1}{cosec\theta} + \dfrac{sec\theta}{cosec^2\theta} + \dfrac{1}{sec\theta} - \dfrac{1}{cosec\theta} - \dfrac{cosec\theta}{sec^2\theta}}$

$= \pink{\sf{\dfrac{sec\theta}{cosec^2\theta} - \dfrac{cosec\theta}{sec^2\theta}}}$

Taking RHS,

$= \pink{\sf{\dfrac{sec\theta}{cosec^2\theta} - \dfrac{cosec\theta}{sec^2\theta}}}$

Hence,

LHS = RHS (Proved)

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Answer 2

Answer:

Science, any system of knowledge that is concerned with the physical world and its phenomena and that entails unbiased observations and systematic experimentation. ... In general, a science involves a pursuit of knowledge covering general truths or the operations of fundamental laws.

Didi why my relevant answers is been deleted?..

and pls talk to me di