Question

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Answer 1

Correct Question

To prove:-

Sin θ (1 + tan θ) + Cos θ (1 + Cot θ ) = Cosec θ + Sec θ

Prove

We know,

★ tan θ = Sin θ/Cos θ .

★ Cot θ = Cos θ/Sin θ.

★ (Sin² θ + Cos² θ) = 1.

★ Sec θ = 1/Cos θ

★ Cosec θ = 1/ Sin θ

Take L.H.S.

➠ Sin θ (1 + tan θ) + Cos θ (1 + Cot θ )

Keep value of tan θ & Cot θ

➠ Sin θ ( 1 + Sin θ/Cos θ ) + Cos θ (1 + Cos θ/Sin θ )

➠ Sin θ ( Sin θ + Cos θ )/(Cos θ) + Cos θ ( Sin θ + Cos θ )/(Sin θ)

Take common (Sin θ + Cos θ)

➠ (Sin θ + Cos θ) [ Sin θ/Cos θ + Cos θ/Sin θ ]

➠ (Sin θ + Cos θ) [ (Sin² θ + Cos² θ)]/(Sin θ .Cos θ)

Keep value of (Sin² θ + Cos² θ)

➠ (Sin θ + Cos θ) [ 1/ (Sin θ.Cos θ) ]

➠ Sin θ/(Sin θ.Cos θ) + Cos θ/(Sin θ.Cos θ)

➠ 1/Cos θ + 1/Sin θ

➠ Cosec θ + tan θ

= R.H.S.

That's proved.

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