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ICSE Class X Math 2015 Solved Paper

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Question : 31
Total: 52
Prove that:
‌
sin‌θ
1−cot‌θ
+‌
cos‌θ
1−tan‌θ
 =cos‌θ+sin‌θ
Solution:  
Given : ‌
sin‌θ
1−cot‌θ
+‌
cos‌θ
1−tan‌θ
 =cos‌θ+sin‌θ
L.H.S. =‌
sin‌θ
1−‌
cos‌θ
sin‌θ
+‌
cos
1−‌
sin‌θ
cos‌θ
‌=‌
sin‌2θ
sin‌θ−cos‌θ
+‌
cos2θ
cos‌θ−sin‌θ
‌=‌
sin‌2θ
sin‌θ−cos‌θ
−‌
cos2θ
sin‌θ−cos‌θ
‌=‌
sin‌2θ−cos2θ
sin‌θ−cos‌θ
‌=‌
(sin‌θ+cos‌θ)(sin‌θ−cos‌θ)
sin‌θ−cos‌θ
‌=sin‌θ+cos‌θ=‌ R.H.S. ‌
L.H.S. = R.H.S.
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