Question

prove that sin theta ( 1+tan theta) + cos theta(1+cos theta) = sec theta + cosec theta

Answer 1

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

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

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

( cos θ + sin θ ) ( tan θ + cot θ )

( cos θ + sin θ ) ( sin² θ + cos²θ ) / ( sin θ cos θ )

( cos θ + sin θ ) / ( sin θ cos θ ) since sin² θ + cos²θ = 1

[ cos θ / ( sin θ cos θ ) ] + [ sin θ / ( sin θ cos θ ) ]

1 / sin θ + 1 / cos θ



cosec θ + sec θ

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Answer 2

Answer:

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

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

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

( cos θ + sin θ ) ( tan θ + cot θ )  

( cos θ + sin θ ) ( sin² θ + cos²θ ) / ( sin θ cos θ )  

( cos θ + sin θ ) / ( sin θ cos θ ) since sin² θ + cos²θ = 1  

[ cos θ / ( sin θ cos θ ) ] + [ sin θ / ( sin θ cos θ ) ]  

1 / sin θ + 1 / cos θ  

cosec θ + sec θ

Hope its help full