Question

show that


√sec²theta+cosec²theta =tan theta +cot theta​

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Answer 1

Step-by-step explanation:

We use the following trigonometric identities:

sec

2

θ=tan

2

θ+1 and

cosec

2

θ=cot

2

θ+1

On adding these, we get:

sec

2

θ+cosec

2

θ=tan

2

θ+cot

2

θ+2

⇒sec

2

θ+cosec

2

θ=tan

2

θ+cot

2

θ+2tanθcotθ=(tanθ+cotθ)

2

⇒

sec

2

θ+cosec

2

θ

=tanθ+cotθ

Hence Proved.

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Answer 2

$\large\underline{\sf{Solution-}}$

Consider LHS

$\rm :\longmapsto\: \sqrt{ {sec}^{2} \theta \: + \: {cosec}^{2} \theta }$

$\rm \:  =  \: \sqrt{(1 + {tan}^{2}\theta) + (1 + {cot}^{2}\theta) }$

$\rm \:  =  \: \sqrt{ {tan}^{2} \theta + {cot}^{2}\theta + 2 }$

can be rewritten as

$\rm \:  =  \: \sqrt{ {tan}^{2} \theta + {cot}^{2}\theta + 2 \times 1 }$

can be further rewritten as

$\rm \:  =  \: \sqrt{ {tan}^{2} \theta + {cot}^{2}\theta + 2 \times tan\theta \times \dfrac{1}{tan\theta } }$

$\rm \:  =  \: \sqrt{ {tan}^{2} \theta + {cot}^{2}\theta + 2 \times tan\theta \times cot\theta}$

$\rm \:  =  \: \sqrt{(tan\theta + cot\theta )^{2} }$

$\rm \:  =  \: tan\theta + cot\theta$

Hence,

$\\ \: \: \: \: \: \: \purple{\boxed{\tt{ \: \: \rm \:  \sqrt{ {sec}^{2} \theta + {cosec}^{2}\theta } =  \: tan\theta + cot\theta \: \: }}}$

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Formula Used

$\boxed{\tt{ {sec}^{2}x - {tan}^{2}x = 1}}$

$\boxed{\tt{ {cosec}^{2}x - {cot}^{2}x = 1}}$

$\boxed{\tt{ {x}^{2} + {y}^{2} + 2xy = {(x + y)}^{2}}}$

$\boxed{\tt{ cotx \: = \: \frac{1}{tanx} \: }}$

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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