online user counteronline user counter

MATHMAA

Only Search Your Site

Trigonometric Problems

1) Prove that \(\tan \Theta +\cot \Theta = \sec \Theta \csc \Theta \)

Solution: 

\(\tan \Theta +\cot \Theta = \sec \Theta \csc \Theta \)
=\(\frac{\sin \Theta }{\cos \Theta }+\frac{\cos \Theta }{\sin \Theta }\)

=\(\frac{sin^{2}\Theta +\cos^{2}\Theta  }{\sin \Theta \cos \Theta }\)

=\(\frac{1  }{\sin \Theta \cos \Theta }\)

=\(\sec \Theta \csc \Theta \)


2)\(\left ( \csc \Theta -1 \right )\left ( \csc \Theta +1 \right )=\cot ^{2}\Theta \)

Solution:

\(\left ( \csc \Theta -1 \right )\left ( \csc \Theta +1 \right )\)

=\(\csc ^{2}\Theta -1\)

=\(\cot ^{2}\Theta \)

3)\(\frac{\tan \Theta -\cos \Theta }{\tan \Theta \cos \Theta }=\sec \Theta -\cot \Theta \)

Solution:

\(\frac{\tan \Theta -\cos \Theta }{\tan \Theta \cos \Theta }\)

=\(\frac{\tan \Theta  }{\tan \Theta \cos \Theta }\)-\(\frac{\cos  \Theta  }{\tan \Theta \cos \Theta }\)

=\(\sec \Theta -\cot \Theta \)

4)\(\cot \alpha \tan \alpha -\sin^{2}\alpha =\cos ^{2}\alpha \)

Solution:

We know that \(\cot \alpha \tan \alpha \)=1

\(\cot \alpha \tan \alpha -\sin ^{2}\alpha \)=\(1-\sin ^{2}\alpha \)

=\(\cos ^{2}\alpha \)


5)\(1-\sec \Theta \cos ^{3}\Theta =\sin ^{2}\Theta \)

Solution:

\(1-\frac{1}{\cos \Theta }\cos ^{3}\Theta \)

=\(1-\cos ^{2}\Theta \)

=\(\sin ^{2}\Theta \)

6)\(\csc \Theta \cos \Theta =\cot \Theta \)

Solution:


\(\csc \Theta \cos \Theta  \)=\(\frac{1}{\sin \Theta }\cos \Theta \)

=\(\cot \Theta \)

7)\(1+\tan ^{2}\left ( -\Theta  \right )=\sec ^{2}\Theta \)

Solution:

We know that \(\tan \left ( -\Theta  \right )=-\tan \Theta \)

\(1+\tan ^{2}\left ( -\Theta  \right )=1+\left ( -\tan \Theta  \right )^{2}\)

=\(1+\tan ^{2}\Theta \)

=\(\sec ^{2}\Theta \)

8)\(\frac{1+\sin \Theta }{1-\sin \Theta }=\frac{\csc \Theta +1}{\csc \Theta -1}\)

Solution:

We know that \(\sin \Theta =\frac{1}{\csc \Theta }\)

\(\frac{1+\sin \Theta }{1-\sin \Theta }\)

=\(\frac{1+\frac{1}{\csc \Theta }}{1-\frac{1}{\csc \Theta } }\)

=\(\frac{\csc \Theta +1}{\csc \Theta -1}\)

9)\(\frac{1-\sin \Theta }{1+\sin \Theta }=\frac{\csc \Theta -1}{\csc \Theta +1}\)

solution: Practice this , it is similar to above.

10)



SHARE YOU ENORMOUS EFFORT AND SMART EXAMPLES HERE

!! NEED MORE HELP !!

SBI! Case Studies