Question

2. Prove the identities: (9 marks) b) sin β +tan β 1+sec β sin

Answer 1

2. Answer:

a):

Note the following formula to prove following identity:

$\csc ^2\left(x\right)-\cot ^2\left(x\right)=1$

$x^2-y^2=\left(x+y\right)\left(x-y\right)$

Take the LHS:

  $\csc ^4\left(x\right)-\cot ^4\left(x\right) =\left(\csc ^2\left(x\right)\right)^2-\left(\cot ^2\left(x\right)\right)^2$

$=\left(\csc ^2\left(x\right)+\cot ^2\left(x\right)\right)\left(\csc ^2\left(x\right)-\cot ^2\left(x\right)\right)$      

$=\left(\csc ^2\left(x\right)+\cot ^2\left(x\right)\right)\left (1)$

$=\left(\csc ^2\left(x\right)+\cot ^2\left(x\right)\right)$

$=RHS$

Hence Proved

b):

Take the LHS: To solve in the editor, We repleced BETA with x. You can replace it with BETA while you note down.

$\frac{\sin \left(x\right)+\tan \left(x\right)}{1+\sec \left(x\right)} = \frac{\sin \left(x\right)+\frac{\sin \left(x\right)}{\cos \left(x\right)}}{1+\sec \left(x\right)}$

$=\frac{\frac{\sin \left(x\right)}{\cos \left(x\right)}+\sin \left(x\right)}{1+\frac{1}{\cos \left(x\right)}}$

$=\frac{\frac{\sin \left(x\right)}{\cos \left(x\right)}+\sin \left(x\right)}{\frac{\cos \left(x\right)+1}{\cos \left(x\right)}}$

$=\frac{\frac{\sin \left(x\right)+\sin \left(x\right)\cos \left(x\right)}{\cos \left(x\right)}}{\frac{\cos \left(x\right)+1}{\cos \left(x\right)}}$

$=\frac{\sin \left(x\right)+\sin \left(x\right)\cos \left(x\right)}{\cos \left(x\right)+1}$

  $=\frac{\sin \left(x\right)\left(1+\cos \left(x\right)\right)}{\cos \left(x\right)+1}$

$=\sin \left(x\right)$

$= RHS$

Hence proved

c):

Take the LHS:

$\frac{\left(\cos ^2\left(x\right)-\sin ^2\left(x\right)\right)}{\cos ^2\left(x\right)+\sin \left(x\right)\cos \left(x\right)} = \frac{\left(\cos \left(x\right)+\sin \left(x\right)\right)\left(\cos \left(x\right)-\sin \left(x\right)\right)}{\cos ^2\left(x\right)+\cos \left(x\right)\sin \left(x\right)}$

$=\frac{\left(\cos \left(x\right)+\sin \left(x\right)\right)\left(\cos \left(x\right)-\sin \left(x\right)\right)}{\cos \left(x\right)\left(\cos \left(x\right)+\sin \left(x\right)\right)}$

  $=\frac{\cos \left(x\right)-\sin \left(x\right)}{\cos \left(x\right)}$

   $=\frac{ \cos \left(x\right)}{\cos \left(x\right)}-\frac{\sin \left(x\right)}{\cos \left(x\right)}$

  

  $=1-\frac{\sin \left(x\right)}{\cos \left(x\right)}$

$=1-tan\left(x\right)$

     $= RHS$

Hence proved