Question

Please do all steps.

Answer 1

Solution

Prove:-

• sin θ( 1 + tan θ) + cos θ (1 + cot θ) = sec θ + cosec θ.

Explanation

Take first Left Hand Side.

➛ sin θ( 1 + tan θ) + cos θ (1 + cot θ)

We Have,

$\:\:\:\:\:\:\small\star{\tt{\red{\:\tan \theta\:=\:\dfrac{\sin \theta}{\cos \theta}}}}$

$\:\:\:\:\:\:\small\star{\tt{\red{\:\cot \theta\:=\:\dfrac{\cos \theta}{\sin \theta}}}}$

➛ sin θ ( 1 + sin θ/cos θ) + cos θ ( 1 + cos θ/ sin θ)

➛ sin θ ( sin θ + cos θ)/cos θ + cos θ ( cos θ + sin θ)/ sin θ

➛[sin² θ( sin θ + cos θ) + cos² θ( sin θ + cos θ)]/(sin θ cos θ)

$\:\:\:\:\:\:\small\star{\tt{\red{\:take\:common\:(\sin \theta\:+\:\cos \theta)}}}$

➛ ( sin θ + cos θ) [(sin² θ + cos² θ) /(cos θ. sin θ)]

We Know,

$\:\:\:\:\:\:\small\star{\tt{\red{\:(\sin^2 \theta\:+\:\cos^2 \theta)\:=\:1}}}$

➛ ( sin θ + cos θ)[ 1/(cos θ. sin θ) ]

➛ sin θ / (cos θ. sin θ) + cos θ /(cos θ. sin θ)

➛ 1/(cos θ) + 1/(sin θ)

We Know,

$\:\:\:\:\:\:\small\star{\tt{\red{\:\dfrac{1}{\cos \theta}\:=\:\sec \theta}}}$

$\:\:\:\:\:\:\small\star{\tt{\red{\:\dfrac{1}{\sin \theta}\:=\:\csc \theta}}}$

➛ sec θ + csc θ

Or,

➛ sec θ + cosec θ

R.H.S.

That's Proved.

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