1
Question
Prove that: cos−16365+2tan−115=sin−135
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Solution
cos−16365+2tan−115=sin−135
Taking L.H.S., sin−1√1−(6365)2+sin−12.151+(15)2.....⎡⎢
⎢
⎢
⎢⎣Using,cos−1x=sin−1√1−x22tan−1x=sin−12x1+x2⎤⎥
⎥
⎥
⎥⎦
=sin−1√1−39694225+sin−12/526/25
=sin−1√4225−39694225+sin−1513
=sin−1√2564225+sin−1513
=sin−11665+sin−1513
[Using sin−1x+sin−1y=sin−1(x{√1−x2}+y{√1−x2})]
=sin−1⎧⎨⎩1665√1−(513)2+513√1−(1665)2⎫⎬⎭
=sin−1(1665×1213+513×6365)
=sin−1(16×12+31513×65)
=sin−1(50765×13)=sin−135
Taking L.H.S., sin−1√1−(6365)2+sin−12.151+(15)2.....⎡⎢ ⎢ ⎢ ⎢⎣Using,cos−1x=sin−1√1−x22tan−1x=sin−12x1+x2⎤⎥ ⎥ ⎥ ⎥⎦
=sin−1√1−39694225+sin−12/526/25
=sin−1√4225−39694225+sin−1513
=sin−1√2564225+sin−1513
=sin−11665+sin−1513
[Using sin−1x+sin−1y=sin−1(x{√1−x2}+y{√1−x2})]
=sin−1⎧⎨⎩1665√1−(513)2+513√1−(1665)2⎫⎬⎭
=sin−1(1665×1213+513×6365)
=sin−1(16×12+31513×65)
=sin−1(50765×13)=sin−135
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