Question

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Prove that (sin alpha +cos alpha) (tan alpha+cot alpha) =sec alpha + cosec alpha​

Answer 1

Step-by-step explanation:

let alpha =a

sin a + cos a(tan a + cot a)

sin a + cos a(sin a/cos a+cos a/ sin a)

sin a+ cos a(sin*a+cos*a/sinacosa)

(sina+cosa)(1/sinacosa)

sina +cosa/sinacosa

(sina/sinacosa)+(cosa/sinacosa)

w k t..1/cos=sec and 1/sin=cosec

1/cosa + 1/sin a

sec a + cosec a

as a = alpha

sec aplha + cosec alpha

hence proved

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Answer 2

Step-by-step explanation:

L.H.S =

$( \sin \alpha + \cos\alpha ) ( \tan \alpha + \cot\alpha ) \\ \\ = ( \sin \alpha + \cos \alpha ) ( \frac{ \sin \alpha }{ \cos \alpha } + \frac{ \cos \alpha }{ \sin \alpha } ) \\ \\ =( \sin \alpha + \cos \alpha ) \frac{( { \sin}^{2} \alpha + { \cos }^{2} \alpha) }{( \sin \alpha \cos \alpha ) } \\ \\ = (\sin \alpha + \cos \alpha ) \times \frac{1}{( \sin \alpha \cos \alpha ) } \\ \\ = \frac{ \sin \alpha }{ (\sin \alpha \cos \alpha ) } + \frac{ \cos \alpha }{ (\sin \alpha \cos \alpha ) } \\ \\ = \frac{1}{ \cos \alpha } + \frac{1}{ \sin \alpha } \\ \\ = \sec \alpha + \csc \alpha$

= R.H.S

°.° L.H.S = R.H.S

Hence, Proved