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Question
Sin (1+ tan ) + cos (1+ cot)= sec + cosec.
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Solution
L.H.S= sin θ ( 1 + tan θ ) + cos θ ( 1 + cot θ ) (Given)
=>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
L.H.S=R.H.S
Hence proved
=>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
L.H.S=R.H.S
Hence proved
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