Question

Show that $(1+{\tan }^2\theta ){\cos }^2\theta =1$

Answer 1

Let usfirstfind the value of left hand side (LHS) that is $(1+{\tan }^2\theta ){\cos }^2\theta$ as shown below:

$(1+{\tan }^2\theta ){\cos }^2\theta ={\sec }^2\theta {\cos }^2\theta (\because {\sec }^{2}x=1+{\tan }^{2}x)=\frac{1}{{\cos }^2\theta }\times {\cos }^2\theta (\because \sec \theta =\frac{1}{\cos \theta })=1=RHS$

Since LHS=RHS,

Hence, $(1+{\tan }^2\theta ){\cos }^2\theta =1$.