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MATHMAA

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.

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